numerical prediction of the hydrodynamic performance of a centrifugal pump in cavitating flows

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERINGCommun. Numer. Meth. Engng 2007; 23:363–384Published online 7 September 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.909

Numerical prediction of the hydrodynamic performanceof a centrifugal pump in cavitating flows

Jun Li1,∗,†, Lijun Liu2,‡ and Zhenping Feng1,§

1Institute of Turbomachinery, School of Energy and Power Engineering, Xi’an Jiaotong University,Xi’an 710049, China

2Research Institute for Applied Mechanics, Kyushu University, Fukuoka 816-8580, Japan

SUMMARY

A computational modelling for the prediction of the hydrodynamic performance of a centrifugal pump incavitating flows is presented in this paper. The cavitation model is implemented in a viscous Reynolds-averaged Navier–Stokes solver. The cavity interface and shape are determined using an iterative procedurematching the cavity surface to a constant pressure boundary. The pressure distribution, as well as itsgradient on the wall, is taken into account in updating the cavity shape iteratively. Numerical validationof the present cavitation model and algorithms is performed on different headform/cylinder bodies for arange of cavitation numbers through comparing with the experimental data. Flow characteristics trendsassociated with off-design flow and twin cavities in the blade channel are observed using the presentedcavitation prediction. The rapid drop in head coefficient at low cavitation number is captured for twodifferent flow coefficients. Local flow field solution illustrates the principle physical mechanisms associatedwith the onset of breakdown. Copyright q 2006 John Wiley & Sons, Ltd.

Received 6 January 2006; Revised 3 July 2006; Accepted 5 July 2006

KEY WORDS: cavitation; Reynolds-averaged Navier–Stokes; liquid/vapour interface tracking method;centrifugal pump; numerical simulation

1. INTRODUCTION

Cavitation is a widely existing hydrodynamic phenomenon that has received much attention over thepast several decades. Cavitation physics plays an important role in the design and operation of many

∗Correspondence to: Jun Li, Institute of Turbomachinery, School of Energy and Power Engineering, Xi’an JiaotongUniversity, Xi’an 710049, China.

†E-mail: [email protected]‡E-mail: [email protected]§E-mail: [email protected]

Contract/grant sponsor: Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP);contract/grant number: 20040698049

Copyright q 2006 John Wiley & Sons, Ltd.

364 J. LI, L. LIU AND Z. FENG

liquid handling turbomachines and devices. In many pump applications, large-scale developed, or‘sheet’ cavities form on the blade and endwall surfaces when the pump operates off-design flowor at low system pressure. In particular, the presence of cavitation in hydraulic machinery can notonly lead to significant flow blockage and changes in flow pattern, which in turn lead to decreasein flow efficiency and massflow capability, degrade hydrodynamic load capability and deterioratemachine performance, but also result in intense erosion of the working wall surfaces, loud noiseand serious vibration, even destroy the machine when cavitation breakdown occurs. Ultimately,cavitation breakdown can occur as characteristics by a very rapid decrease in impeller head risecoefficient [1]. For these reasons, it is of interest to the pump designer to be able to model large-scale sheet cavitation toward understanding the physics of, designing away from, and designingto accommodate the effects of cavitation.

Numerical simulation of sheet cavitating flows poses unique challenge, both in modelling of thephysics and in developing robust numerical methodology. The major difficulty arises due to thelarge density changes associated with phase change. Furthermore, the location, extent and shapeof the sheet cavitation are strongly dependent on the pressure field, which in turn is influenced bythe flow geometry and conditions [2]. The classical approach of modelling the sheet cavitation isto treat this liquid/vapour interface as a streamline with constant pressure and velocity potentialequations are employed to describe the flow field outside the vapour bubble. The shape and size ofthe cavity are determined from dynamic equilibrium assumptions across the interface, with bubblelength and closure conditions provided. Different adaptations of boundary element methods basedon potential flow theory remain in widespread use nowadays due to their inherent computationalefficiency.

Recently, with the advent of inexpensive powerful computers and urgent request for under-standing the realistic physics of cavitation, more general computational fluid dynamics (CFD)approaches have been developed and they are seeing more and more use in predicting and evaluat-ing flow fields of this nature in different liquid handling machines and situations [3–20]. Deshpandeet al. [3, 4] developed an Euler analysis method and then extended it by solving the Reynolds-averaged Navier–Stokes equations (RANS) to predict the geometrical characteristics of cavitationbubble. Chen and Heister [5, 6] reported some effective numerical techniques for the sheet cav-itation. Hirschi et al. [7] investigated the hydrodynamic performance of a centrifugal pump incavitating flows using single-phase Navier–Stokes solver and cavitation model. In his cavitationmodel, the pressure gradient at the liquid/vapour interface was not taken into account. The initialcavitation shape is assumed in his computation. Kunz et al. [10, 12] presented a multiphase CFDmethod to analyse the sheet cavitating flows of the centrifugal pump in cavitating flows. Thedifferential model employed was the homogeneous two-phase RANS equations, wherein mixturemomentum and volume continuity equations were solved along with vapour volume fraction conti-nuity. Mass transfer modelling was provided for the phase change associated with sheet cavitation.Senocak et al. [15, 16] and Wu et al. [17, 18] developed a new interfacial dynamics-based cavitationmodel coupled with the Navier–Stokes solutions to predict the turbulent cavitating flows. Wienkenet al. [19] presented an approach to apply the large-eddy simulation for the prediction of cavitationinception. The above works has been performed on sheet cavitating flows by RANS method-ologies. The most attractive features of RANS methodologies are that many realistic physicsof cavitation can be readily incorporated and modelled, and, they are feasible for applicationin very complicated configurations. The wall detachment point and bubble length are naturallydetermined from the computation, unlike the potential model in which they are usually providedas a priori knowledge.

Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2007; 23:363–384DOI: 10.1002/cnm

NUMERICAL PREDICTION OF THE HYDRODYNAMIC PERFORMANCE 365

The adaptations of RANS methodologies for modelling sheet cavitating flows can be groupedinto two distinctly different approaches. One approach, which is often referred to as interfacecapturing method, treats the flow as one compressible continuum fluid in the whole flow fieldbasing on the idea of two-phase flow model, with variable properties and a pseudo-density whichwidely varies between liquid and vapour extremes. The most attractive features of this approachare that no ad hoc wake closure model is required and it can treat travelling cavitations as well assheet cavitations.

This approach remains a number of numerical and physical modelling challenges. Since theliquid/vapour interface is very thin and the gradients of fluid properties are very large at theinterface, very high spatial resolution, very accurate and delicate differential scheme must beadapted in the vicinity of the interface, which is constantly changing during the iteration procedureor developing with time. Therefore, large size of mesh and vast computation resources are required.In addition, the greatest challenge in the implementation of the interface capturing methods isin deriving a suitable constitutive relationship between the pseudo-density and the other flowvariables, such as the pressure, in order to close the problem statement in this approach. Theformulations of this relationship proposed in the existing literatures [8–10, 12, 13] so far are moreempirical.

Another distinct class of numerical methods, hereafter referred to as interface tracking approach,seeks a solution in the liquid domain along with a description of the boundary of the cavitatingarea. The liquid/vapour interface is treated as part of the boundary of the computation domainand it is located by assuming that the cavitating region is at a constant pressure equal to thelocal vapour pressure. The flow field is solved by a single-phase flow solver. Because of thenon-linear relationship and interaction between the vapour cavity shape and the external flowfield of the liquid, the solution of interface is obtained by an iteration procedure. The interfaceis updated iteratively until a unique convergent shape is achieved with constant pressure alongit. Numerically, however, the interface tracking approach has to employ a wake closure modelto approximate the two-phase behaviour in the wake region at the end of the cavitation region,since it is impossible to impose a constant pressure condition on the entire cavity surface whilethe recovery of pressure occurs at the aft end of the cavitation [11]. Despite that it is difficultto introduce physics into the wake closure model and there is no general way to define it, mostauthors [6, 7, 14], fortunately, agree that the choice of the wake closure model, within limits, haslittle influence to the forebody of the cavity and their researches have proved this assertion. Themost salient feature of the interface tracking approach is that it has bypassed all those numericaldifficulties of the interface capturing approach and physical modelling difficulties of the two-phase flow models while retaining the capabilities to capture most features of the viscous sheetcavitating flows.

This paper is to implement the interface tracking approach with a turbulent viscous incom-pressible RANS flow solver for the prediction of the hydrodynamic performance of a centrifu-gal pump in sheet cavitating flows. The cavity interface and shape are determined using aniterative procedure matching the cavity surface to a constant pressure boundary. The pressuredistribution, as well as its gradient on the wall, is taken into account in updating thecavity shape iteratively. The accuracy and stability of the cavitation model and algorithmsis to be well validated by performing for different headform/cylinder bodies at differentcavitation numbers. The numerical results of the analyses for a centrifugal pump withthree blade impellers operating across a range of flow coefficient and cavitation numbers arepresented.



2. CAVITATION MODEL AND ALGORITHMS

2.1. RANS

The cavitation model is implemented in a RANS viscous flow solver. The mass conservationequation and momentum equations of the fluid flow can be written as follows in Cartesianco-ordinates for steady, incompressible viscous flows:

�(�ui )

�xi= 0 (1)

�(�u jui )

�x j= − �p

�xi+ ��i j

�x j+ bi (2)

�i j = �eff

(�ui�x j

+ �u j

�xi

)− 2

3�i j�eff

�ul�xl

�eff = � + �t

The Baldwin–Lomax turbulence model [21] is employed in the computations to approximatelyevaluate the turbulence viscosity because of its simplicity and higher computational efficiencycompared with one- or two-equation models.

�t =

⎧⎪⎪⎨⎪⎪⎩0.16�y2[1 − exp(−y+/A+)]2|�|, 0�y�y′

0.02688�Fw

[1 + 5.5

(0.3y

ymax

)6]−1

, y�y′

⎫⎪⎪⎬⎪⎪⎭ (3)

where y denotes the distance away from the wall. Range from 0 to y′ is the wall region, where y′represents the smallest value of y at which the values of �t in the inner and outer layer are equalto each other.

y+ =√

�w�w

�w

y, A+ = 26 (4)

where �w and �w are the shear stress and molecular viscosity at the wall. � is the strength of flowvortex. And

Fw ={0.25ymax(Umax −Umin)

2

Fmax

}or Fw = ymaxFmax (5)

The smaller value of Fw is taken. Here, Fmax is the maximal value of F(y) defined by

F(y)= y|�|[1 − exp(−y+/A+)] (6)

And ymax is the value of y at that point. The value of Umin = 0 is for the boundary layer.Using the technique of non-staggered grid, a pressure-based algorithm based on the so-called

SIMPLEC method for general curvilinear co-ordinates is adopted to couple the momentum equa-tions and the continuity equation in the flow computations. The covariant velocity projectionsrather than the Cartesian velocity components are selected as the dependent variables, resulting in



the cross-pressure gradient terms in the momentum equations disappear. The pressure correction,rather than the pressure itself, is derived and solved in this method. A momentum interpolationscheme, which was newly developed to suppress the non-physical pressure oscillation, is used forgetting the flow flux through the surfaces of the control volume. The discretization of the govern-ing equations is performed using the finite volume approach. The convective and the orthogonaldiffusion terms in the momentum equations are treated by using the two-order upwind differencingscheme. The non-orthogonal diffusion terms are evaluated by using the central differencing schemeand lumped into the source term of the discretization equations. The resulting discretization equa-tions are solved iteratively by the successive line over-relaxation method (SLOR), with ADI andblock additive correction technique to speed up the convergence speed [14].

Neumann boundary condition is imposed for pressure correction on all boundaries except forthe period boundaries if applicable, where the period condition is enforced for all variables. Theinlet pressure level is prescribed in the computation. A uniform velocity field is defined at theinlet boundary and Neumann condition is specified at the outlet boundary. The non-slip conditionis applied on all wall surfaces. Since the cavity surface is treated as a free streamline, the freesurface boundary condition and impermeability condition are enforced on the cavity surface, thatis, �Vt/�n = 0 and Vn = 0, where n is the local normal direction and t is the local tangentialdirection to the cavity surface.

2.2. Cavity shape updating scheme

The sheet cavitation is modelled as a large cavity bubble on the wall with an interface to the bulkflow. There are three assumptions in this cavitation model.

(1) The liquid/vapour interface is a free surface;(2) The pressure inside the cavity equals the local vapour pressure of the fluid;(3) The rear part of the cavitation bubble is approximated with a wake model, where the pressure

is no longer equal to the vapour pressure.

Figure 1 shows a schematic description of a cavitation bubble on a headform/cylinder body,where A is the inception point, B the point linking the forebody and afterbody (wake region) ofthe cavity and C the end point of cavitation. There is three highlights need to pay attention toobtain the cavity shape: the location of the inception point A, the profile of the forebody AB andthe profile of the afterbody BC .

main flow

Cavitation bubble

Figure 1. A sheet cavitation bubble on a headform/cylinder body.



The cavitation model has been implemented in a viscous RANS solver, in which a pressure-based SIMPLEC algorithm is adopted. The computation method of cavity modelling consists ofassuming the cavity interface as a free surface boundary of the computation domain and computingthe single-phase flow. Based on the information of the flow field, a cavity updating scheme is usedto update the interface shape in order to reach a constant pressure equal to the vapour pressure onit. As the cavity shape has an influence on the mean flow, an iterative process is applied betweenthe flow field computation and the cavity surface updating. The cavitation inception point andbubble length are automatically determined in this iterative procedure. A priori knowledge of thecavity shape, therefore, is not required.

The calculation procedure can be outlined in three steps: First, a cavitation-free flow computa-tion is performed. According to the information given by the flow computation, the inception andclosure point of cavitation are located. The thickness of the cavity is assumed to be zero. Then, theflow computation is repeated, enforcing the free surface condition: the normal velocity componentand the shear rate tensor on the cavity surface to be zero, on the cavity surface. Finally, according tothe flow computation, the cavity shape is updated by the interface-updating scheme. Meanwhile, theinception and closure point are checked and renewed. The mesh for the flow domain is then adaptedto the new cavity interface. And, the previous solution of the flow computation is extrapolated onthe new mesh, being taken as the initial field for the flow computation of the next iteration. The pro-cess of step two and three is repeated until the pressure distribution along the cavity surface reachesto a constant pressure equal to the vapour pressure in a prescribed permitted error. While the cavitysurface is constantly updated from one step to the next one, the inception and enclosure points arealso checked and renewed at each step. The inception point is always located at the point on the wallwhere the pressure is minimization. The enclosure domain of cavitation is modelled by a cubic poly-nomial function curve smoothly linking the cavity surface and the wall. The length of the enclosuredomain is empirically specified. The attached point of the cavity is then automatically located.

Regarding to the two-phase turbulent wake region, since we are still not aware of the flowstructure and mechanisms in it, it is modelled by an afterbody defined by a cubic polynomialfunction curve for the sake of simplicity, as most authors did in their works [7, 11, 14]. The mostturbulence-dominated region is the wake region at the rear part of the cavitation where there arerecirculation and wall jet-like flow structure. It is two-phase and strongly time dependent. Themechanisms of this kind of two-phase flows are so extremely complex that, up to now, no availableturbulence model is capable of treating this kind of flow properly [6]. Therefore, in our model,this area is modelled with a wake closure model. In such a way, the modelling difficulty of thishighly turbulence-dominated time-dependent two-phase flow region is avoided. The flow structureof vapour flow in the cavitation bubble does not need to be solved in our model. Considering thesheet cavitation is very thin and the wake region is modelled with an enclosure model, the flowstructure close to the liquid/vapour interface and solid walls are similar to a simple layer flow.The turbulence effect of the bulk flow is therefore approximated modelled by the Baldwin–Lomaxturbulence model that is efficient and extensively used. The afterbody begins at the point on thecavity surface where the local height is, for example, half of the maximum height of the cavity. Itsmoothly links the forebody of the cavity surface and the local wall surface. The first as well asthe second derivative of the cavity profile at the linking point B is enforced to be continuous. Thelength of the closure domain, as well as the attached point of the cavity, is thereby automaticallydetermined by the wake closure model. However, at the early stage of iteration, since the cavityshape is far from well established, the length of the closure domain calculated from the wakeclosure model may be too long or too short. This may result in both the inner loop iteration of



the flow field calculation and the outer loop iteration of the cavity shape updating time-consumingand instable. In order to avoid this numerical problem of the model, the length of the wake regionis specified associating with the maximal height of the cavity and only the first derivation of thecavity profile at the linking point B is enforced to be continuous. The computations show that thistechnique tends to speed up the iteration procedures of both the inner loop and outer loop.

Hirschi et al. [7] proposed a relationship between the flow field and the shape of the cavity,similar to the following, to update the cavity surface during iteration:

dr = �(pv − p) (7)

where dr is the local change of the cavity thickness. Chen and Heister [6] presented another similarformula for this purpose in their work as follows:

d� =C(pv − p) (8)

where d� is the local change of the inclination angle of the cavity surface and C is an empiricalconstant. Although satisfactory results were got in their works by using the above relationships, thefactor or empirical constant in the above formulas must be very small, in order to avoid numericalinstability and flow field oscillation [6, 7]. Thus, the computation is very time-consuming. In orderto save computation time, an initial cavity shape is established empirically in the work of Hirschiet al. [7]. Analysing the physical mechanism of the flow past around a cavity surface, where thederivative of the cavity thickness and its curvature are continuous on most of the cavity surfacebut discontinuous at the inception point, here is proposed the following equation for updating thecavity surface:

��(n) =∫ s

sb

⎛⎝�1 sgn(pv − p(n))

√|p(n) − pv|‖p(0) − pv‖

⎞⎠ ds

+∫ s

sb

⎛⎝�2 sgn

(�p(n)

�s

)√|�p(n)/�s|‖�p(0)/�s‖

⎞⎠ ds (9)

where ‖p(0) − pv‖=√∫ se

sb(p(0) − pv)2 ds/(se − sb) and ‖�p(0)/�s‖ is defined in the same way.

sb and se are the locations of the cavitation inception and enclosure point, respectively. Superscript ncorresponds to the calculated value of the present iteration. As can be seen, the pressure distributionand its gradient on the wall are taken into account for updating of the cavity shape, whichshould be superior to the existing schemes mentioned above, in which only pressure distributionwas considered. Assuming �1 = �2 = �, formulation (9) can be rewritten as follows after makingderivation calculus on its both sides:

��(n) = �

180�

⎡⎣sgn(pv − p(n))

√|p(n) − pv|‖p(0) − pv‖

⎤⎦+ �

⎡⎣sgn

(�p(n)

�s

)√|�p(n)/�s|‖�p(0)/�s‖

⎤⎦ (10)

This is the formulation of the newly developed cavity shape updating scheme used in this work.The iteration process is terminated when the solution of the flow field has converged and the

following condition gets satisfied:

Dp/Dp0 + = ‖p(n)(s) − pv‖‖p(0)(s) − pv‖�� (11)



where p(s) is the pressure on the forebody of the cavity, � presents the prescribed permissionerror. ‖p(s) − pv‖ is the second modulus of pressure difference over the forebody of the cavitysurface. � is a value between 5× 10−3 and 3× 10−2 in this computation.

3. NUMERICAL VALIDATION

3.1. Grid independent

In this work, an algebraic method is chosen for mesh generation due to its computational effi-ciency. Exponential stretching is used in high-gradient regions. A typical mesh is shown for anaxisymmetric hemispheric headform/cylinder body in Figure 2(a), in which the grid is clusterednear the centreline/wall surface and in the vicinity of the hemispheric headform. The initial mesh(shown in Figure 2(a)) generated for the cavitation-free flow computation is save as the backgroundmesh. All the operations of mesh adaptation and data transferring between the previous and thenext initial solution of the flow computation are performed referring to the background mesh in

(a)

(b)

Figure 2. Mesh for the headform/cylinder body: (a) mesh for cavitation-free flow computation;and (b) local mesh distribution with cavitation.



Figure 3. The computational grid for ogival/cylinder body.

S*

Cp

-2 0 2 4 6 8-1.00

-0.50

0.00

0.50

1.00 grid number 62*181grid number 46*131grid number 32*91experimental data

Figure 4. Pressure coefficient distribution for ogival/cylinder with different grid numbers.

the iteration procedure. After every time the cavity shape has updated, the new profile of the cavityshape is recorded on the background mesh frame and the mesh is adapted to the new flow domainreferring to the background mesh. Figure 2(b) gives a typical local mesh distribution re-grided toadapt to the cavity shape establishment on the wall of a hemispherical headform/cylinder body.The grid is clustered close to the body wall and headform where gradient is usually large. Goodgrid quality is ensured close to the headform regarding to the grid uniformity and skewness.



Before each computation, it is a routine to perform a mesh dependency test in order to pursuea solution independent of the mesh size employed. The present cavitation model is applied andassessed for cavitating flow over ogival/cylinder body with different grid numbers. Experimentaldata are reported by Rouse and McNown [22]. The experiment is conducted in a water tunnelwith cylindrical test objects 0.025m in diameter and 0.2038m in length. The curvature radium ofthe ogive headform profile is two times the radium of the afterbody cylinder. Two-dimensionalaxisymmetric computational grid is built for this problem. The computational grid is shown inFigure 3 with a grid resolution of 46× 131. Figure 4 shows the mesh dependency test of thecavitation prediction for an ogive/cylinder body. The cavitation index is K = 0.24. The curvatureradium of the ogive headform profile is two times the radium of the afterbody cylinder [22].The boundary conditions at the centreline are standard axisymmetric conditions and the far fieldcondition is applied at the inlet, outlet and outer boundaries. The non-slip condition is imposedon the wall and the free surface condition is enforced on the cavity surface. Given in Figure 3 arethe pressure coefficient Cp obtained by three sets of mesh: 32× 91, 46× 131 and 62 × 181. Thenumerical results show that the solution is independent of the mesh size of the presented cavitationmodel and algorithm.

Figure 5 presents the pressure coefficient distributions and cavity shape for the case of anogive/cylinder body for K = 0.32, 0.24, with result of the free cavitation computations. Theexperimental data of Rouse and McNown [22] are also presented in Figure 5. The results inFigure 5 indicate that the location of inception point is accurately predicted for cavitations on thesmooth surface. These results validate the assumed inception criteria and treatments for flows ofthis nature. The strong influence of cavitation number on the pressure distribution and cavity shapeis again highlighted in Figure 5.

S*

Cp

0 2 4-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Cavitation FreePresented Methodology(K=0.24)Presented Methodology(K=0.32)Experimental Data(K=0.24)Experimental Data(K=0.32)

Figure 5. Results for ogival/cylinder body.



0.5 1.5 21 2.5

Z*

0.2

0.4

0.6

0.8

1

1.2

R*

ComputationExperiment

Conic HeadformK = 0.40

r

z

Cavity Wall

Figure 6. Comparison of the calculated cavity shape with experiment data forcone/cylinder body form (K = 0.40).

50 100 150 200 250 300ITER

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

Lo

g10

Dp

/Dp

0

λ=0.2λ=0.5λ=1.0λ=2.0

Figure 7. The iteration history of the cavity updating scheme.

The cavitation flow across the cone/cylinder is used to investigate the flow details of the cavi-tating flow along the cone/cylinder body. A local enlargement plot of the cavity shape comparisonbetween the computation and the experiment [22] is given in Figure 6 for K = 0.40. As can beseen from Figure 6, the cavity shape is well duplicated with the present model as compared withthe experimental results. Even the pressure recovery in the wake region is well predicted.

3.2. Influence of the relaxation factor

In order to test the influence of the relaxation factor � for the present cavitation treatment, differentrelaxation factors � are used to simulate the flow across the hemisphere/cylinder body. These



experimental data are also reported by Rouse and McNown [22]. The same boundary condition isapplied as the ogival/cylinder body case. Figure 7 shows the iteration history of computations atthe flow condition of K = 0.40. In Figure 7 we can see that the convergence speed tends to speed upwith larger value of the relaxation factor. However, the reachable convergence precision decreases(value increases) with the increasing value of relaxation factor. For example, the convergencecriteria prescribed in this work (log10 Dp/Dp0 = −2.0) can be achieved by 214 iterations ofcavity shape updating for � = 0.2, while only half iteration times is needed for � = 0.5. For� = 2.0, only 33 iteration times is needed for this scheme to reach a convergence precision aroundlog10 Dp/Dp0 =−1.4, but this becomes the limit of the convergence precision it can achieve.Therefore, there is a compromise between the convergence precision and computational efficiencywhen considering the selection of the relaxation factor value. In the foregoing computations, avalue of � = 0.5 is chosen to ensure the iteration converge satisfactorily. A local small but suddenoscillation on the curve of iteration history, as marked in Figure 7, is due to the transition of thedefining method of the wake closure model when the iteration approaches to convergence. We cansee from Figure 7 that, the iteration seems to speed up after the transition of the defining methodof the wake closure model since the profile becomes smoother. This verifies the effectiveness ofthe developed wake closure model.

Figure 8 presents the pressure coefficient distribution, cavity thickness distribution and cavityshape for a hemisphere/cylinder body for K = 0.40, 0.30, 0.20 with � = 0.5, respectively. Theresults of the free cavitation computation and the experimental data [22] are included for compar-ison. Results in Figure 8 indicate that the location of inception point is accurately predicted forcavitation. These results validate the assumed inception criteria and treatments for flows of this

S*

Cp

-1 0 1 2 3 4 5 6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2Cavitation FreePresented Methodology(K=0.20)Presented Methodology(K=0.30)Presented Methodology(K=0.40)Experimental Data(K=0.20)Experimental Data(K=0.30)Experimental Data(K=0.40)

Figure 8. Results for hemisphere/cylinder body.



nature. The accuracy and effectiveness of the current cavitation model and algorithms are furtherdemonstrated in Figure 8 for a range of cavitation numbers by comparisons with experimental data.

4. PERFORMANCE PREDICTION OF THE CENTRIFUGAL PUMP

Numerical simulation is performed to predict the cavitation in a centrifugal pump impeller usingthe presented cavitation model and algorithm in this section. The cavitation behaviour and its effecton pump performance are briefly studied. Some interesting phenomena of the sheet cavitation inthe pump impeller are illustrated too.

The geometry of the calculated centrifugal pump impeller is described as follows:

D2 = 348.5, b2 = 18, �2A = 12◦, D1 = 183

blade number NZ = 3, rotational speed n = 800 rpm.The profile of the blade, the computation domain and mesh are given in Figure 9, in which

a typical converged cavity shape is illustrated. The series of computation are performed in twogroups: one group with varied flow rate coefficient but fixed cavitation number, and, another groupwith varied cavitation number but fixed flow rate coefficient.

For the group of flow computations with varied flow rate, there are two series of computation areperformed: one without cavitation for studying the performance of the impeller free of cavitation;another for = 0.07 for studying the response of cavitation behaviour to the flow rate change andcavitation effect on the performance drop of the impeller. Fifteen operating points are calculatedfor the computations without cavitation, and, nine operating points are calculated for the caseof = 0.07. All these operating points are shown and compared in Figure 10, from whichwe can see the performance drop due to cavitation and cavitation breakdown at = 0.066. Figure 11

Cavitation

Figure 9. The computational grid with sheet cavitation bubble of the centrifugal pump impeller.



0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1φ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

ψ

cavitation free

σ = 0.07

Figure 10. The hydrodynamic performance with different flow coefficient.

0.03 0.04 0.05 0.06 0.07φ

0

1

2

3

4

5

6

7

8

0

10

20

30

40

50

60

70

80

leng

th /

LB

(%)

heig

ht /

D2

(%)

0

2

4

6

8

10

12

14

16

volu

me

frac

tion

(%)

Height max.Height meanBubble lengthBubble volume

(σ = 0.07, n = 800 rpm)

Figure 11. Response of cavity size to flow coefficient change.

presents the response of cavitation bubble size to the change of flow rate. The cavity shape, locationand pressure coefficient distribution in the flow channel and that along the blade surfaces for tworepresentative operating points, namely = 0.035 and 0.066, on the ∼� performance curve of = 0.07 are shown in Figures 12 and 13, respectively.



In Figure 11 are the parameters of cavity versus flow coefficient. The bubble height is normalizedby the outlet diameter of the impeller D2. The maximum height is defined as the first modulo ofthe bubble height function over its forebody and, correspondingly, the mean height is defined asthe second modulo. The bubble length is normalized by the length of blade. The bubble volumefraction is the bubble volume normalized by the volume of blade channel. As can be seen fromthis figure, with the increasing of the flow coefficient, the height as well as the volume fractionof the bubble decreases constantly until the operating point = 0.065. Meanwhile, opposite tothe change of bubble height, the bubble length is prone to increase with the increasing of flow

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1S / L

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Cp

cavitation free

σ = 0.07

(a)

7

16

22

28

4

Level Cp

31 1.0028 0.8925 0.7822 0.6719 0.5616 0.4513 0.3410 0.237 0.124 0.011 -0.10

Cavitation

(b)

Figure 12. Results for = 0.035 and = 0.07: (a) pressure distribution along the blade and cavity surface;and (b) cavity shape and pressure contours in the blade channel.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1S / L

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Cp

cavitation free

σ = 0.07

(a)

19

4

25

10

Level Cp

31 1.0028 0.8925 0.7822 0.6719 0.5616 0.4513 0.3410 0.237 0.124 0.011 -0.10

Cavitation

(b)

Figure 13. Results for = 0.066 and = 0.07: (a) pressure distribution along the blade and cavity surface;and (b) cavity shape and pressure contours in the blade channel.

coefficient. From Figures 12(b) and 13(b), it can be seen that the ‘stout’ cavity attached on thesuction surface at low flow rate becomes ‘slender’ with the flow rate increasing. At the sametime, the inception point of cavitation moves backward along the suction surface of the blade.For the flow rate higher than = 0.065, the cavitation develops downstream across the ‘throat’ ofthe blade channel. As we can know from Figures 12(a) and 13(a), in which the abscissa S/L is therelative distance along the blade surface away from the inlet boundary, for flows free of cavitationthe minimum pressure on the suction surface of blade decreases while the pressure gradienton it becomes steeper when the flow coefficient decreases. This changes tendency of pressuredistribution on the blade surface may prevent the cavitation from developing downstream alongthe blade surface but makes it grow in the normal direction when flow coefficient decreases. From



the comparison of pressure distribution between flows with and without cavitation, the effects ofcavitation on the flow pattern are highlighted in these figures.

When the flow coefficient approaches to = 0.066, the behaviour of the cavity to the flowcoefficient changes dramatically. With flow coefficient increasing, the bubble height as well asthe bubble volume fraction turns back to increase suddenly. And, particularly, the bubble lengthincreases dramatically. Corresponding to this interesting phenomenon shown in Figure 13, we cannotice in Figure 12 that, while the performance drop due to cavitation has no dramatic changebetween = 0.035 and 0.065, the head rise coefficient suddenly drops down at the working pointof = 0.066, which is indicated in these figures. Therefore, it may be interesting to analyse theflow field under this flow condition. Different from those results obtained for the flow coefficientsless than = 0.065, it is found in Figure 13 that, not only the flow on the suction side, butalso that on the pressure side of the blade, has been cavitated. And, both cavities on the bladesurface develop backward across the ‘throat’ of the blade channel. The similar numerical resultwas published by Medvitz et al. [12] using two-phase flow model. Figure 14 shows the two

Cavitation bubble

Figure 14. Two attached cavitation bubbles along the centrifugal impeller.

0.05 0.1 0.15 0.2σ

0.7

0.8

0.9

1

1.1

1.2

ψ

φ = 0.047

φ = 0.058

Figure 15. Head rise coefficient versus cavitation number.



sheet cavitation bubble along the blade using numerical simulation [12]. This point can be able todemonstrate the reliability of the presented numerical results.

In order to study the response of cavitation behaviour to cavitation number and its effect onimpeller performance at different flow coefficients, two series of computation are performed inanother group of computation: one for = 0.058, in which seven operating points with cavitationnumber ranging from = 0.064 to 0.135 (cavitation free); another for = 0.047, in which nineoperating points with cavitation number ranging from = 0.0525 to 0.185 (cavitation free). Allthese operating points are presented and compared in Figure 15. As can be seen, for both flow

0.05 0.075 0.1 0.125 0.15σ

0

1

2

3

4

5

6

7

8

heig

ht /

D2

(%)

heig

ht /

D2

(%)

0

10

20

30

40

50

60

70

80

leng

th /

LB

(%

)

0

2

4

6

8

10

12

14

16

volu

me

frac

tion

(%)

0

10

20

30

40

50

60

70

80

leng

th /

LB

(%

)

0

2

4

6

8

10

12

14

16

volu

me

frac

tion

(%)



(φ = 0.058, n = 800 rpm)

(a)

0.05 0.075 0.1 0.125 0.15 0.175 0.2

σ

0

1

2

3

4

5

6

7

8(φ = 0.047, n = 800 rpm)

(b)

Figure 16. Response of cavitation behaviour to cavitation number: (a) = 0.058; and (b) = 0.047.



coefficients the head rise coefficient has a decreasing tendency with the reduction of cavitationnumber. For a certain flow coefficient, there is a critical operating point on its ∼� curve. The headrise coefficient drops down abruptly when the cavitation number gets less than that critical value. For = 0.058 and 0.047 in the present computations, the critical cavitation number corresponds to thelast calculated operating point on their performance curve, i.e. = 0.064 and 0.0525, respectively.It is obviously that the critical cavitation number for = 0.058 is larger than that for = 0.047.It means that the performance of machine is prone to corrupt at higher flow coefficient, which isconsistent with the conclusion we got from Figure 10.

Given in Figure 16 are the relationships of the parameters of bubble size versus cavitationnumber for both flow coefficients. For each case, all the geometrical parameters of the cavity, i.e.the bubble length, height and volume fraction have a increasing tendency with the reduction ofcavitation number. When approaching to the critical cavitation number, the growth of cavity speedsup quickly. Comparing the size of the cavity for both flow coefficients, the magnitudes of bubblelength are approximately the same order. But the values of bubble height and volume fraction for = 0.047 are more than two times the values of that for = 0.058.

-0.03

0.24

0.35

0.47

0.09

0.650.80

0.88

Impeller

Cavity

Impeller

Cavity

0.30

0.41

0.54

0.850.68

0.02

-0.03

(a) (b)

0.28

0.510.59 0.67 0.75

-0.03

0.01

0.83

Impeller

Cavity

(c)

Figure 17. Pressure contours and cavity shape in the blade channel: (a) =0.047 and = 0.09;(b) = 0.047 and = 0.057; and (c) = 0.047; and = 0.0525.



A comparison of cavity shape and pressure distribution along blade surface is made in Figure 17for a few representative cavitation numbers with = 0.047. It is shown that, with cavitation numberdecreasing, the cavity not only grows in the normal direction to the blade surface, but also at thesame time develops toward downstream quickly. With the cavitation number approaching to itscritical value, the cavity develops downstream across the ‘throat’ of the blade channel and apparentlychanges the flow pattern throughout the flow channel. By noting the pressure distribution presentedin the figure for = 0.0525, the cavitation tends to occur on the pressure side of the blade, too.And, at this situation, the cavitation is prone to change the flow pattern in the flow channeldramatically comparing to the free cavitation flow field. These phenomena are reproduced for thecase of = 0.058. We can, therefore, expect that, twin cavities’ cavitation is prone to occur whenthe cavitation number gets small enough, even for low flow rate situations. The twin cavities onboth sides of the blade, together with the resultant reverse flow in the channel, may dramaticallychange the flow pattern throughout the flow channel and then deteriorate the impeller performance.

5. CONCLUSIONS

An improved cavitation model and some original treatments have been developed for the predictionof attached cavitation. Particular emphasis is placed on unique aspects of the numerical treatmentsof cavitation modelling including the location of inception point, the wake closure model and,particularly, the cavity shape updating scheme. The cavitation model is implemented in a viscousNavier–Stokes flow solver with turbulence model employed. A new cavity shape updating schemeis derived, in which the pressure difference as well as the pressure gradient is taken into account.The cavitation model and treatments are validated against a series of experiments. The accuracyof the cavitation model and algorithm are demonstrated by comparisons of the predicted resultswith available experimental data.

The capability to capture some important phenomena in complex flows is shown by the cavitationprediction in a centrifugal pump impeller. The twin cavities’ cavitation is well reproduced in thecomputation, showing its capability to deal with cavitating flows resulting from multiple cavitationregions. The cavitation behaviour and its effects on the impeller performance are studied for arange of cavitation numbers and flow coefficients.

NOMENCLATURE

b2 the blade width at the impeller outletC coefficient factors in formulaCp pressure coefficient (Cp = (p − p∞)/(p0 − p∞))

D1, D2 inlet and outlet diameter of the impellerH head riseK cavitation index (K = (p∞ − pv)/(p0 − p∞))

L the total length of the impeller channeln rotation speed of the impellerNZ blade number of the impellerNPSH net positive suction headp static pressure



pv vapour pressureQ flow rater local thickness of the cavity surfaceS, s distance along streamline or wall surfaceu2 the tip speed of the impeller bladeVn velocity component normal to the cavity surfaceVt velocity component tangential to the cavity surfaceZ , R co-ordinates in cylinder frame� local slope angle of the cavity surface�2A the blade angle at the impeller outlet� relaxation factor in cavity shape updating scheme� molecular viscosity�t turbulence viscosity�eff effective viscosity� density of the fluid cavitation number (=NPSH/(u22/2g)) flow rate coefficient (= Q/(�D2b2u2))� head rise coefficient (�= H/(u22/2g))� vortex strength of the flow

ACKNOWLEDGEMENTS

This work was supported by the grants from Specialized Research Fund for the Doctoral Program ofHigher Education (SRFDP) (20040698049). The authors would like to extend his appreciation to Prof. H.Tsukamoto of Kyushu Institute of Technology, Japan and Dr T. Tanaka of Yatsushiro National Collegeof Technology, Japan, for their helpful discussions.

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numerical prediction of the hydrodynamic performance of a centrifugal pump in cavitating flows

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