Study on Flow Characteristics in Volute of …downloads.hindawi.com/journals/mpe/2019/2567659.pdfcentrifugal pump for the low ow-rate (. d) condition. Obviously, the main frequency

Research ArticleStudy on Flow Characteristics in Volute of Centrifugal PumpBased on Dynamic Mode Decomposition

Yi-bin Li Chang-hong He and Jian-zhong Li

College of Energy and Power Engineering Lanzhou University of Technology Lanzhou 730050 China

Correspondence should be addressed to Yi-bin Li liyibin58163com

Received 31 January 2019 Accepted 4 April 2019 Published 16 April 2019

Academic Editor Sergey A Suslov

Copyright copy 2019 Yi-bin Li et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To investigate the unsteady flow characteristics and their influencemechanism in the volute of centrifugal pump the Reynolds time-averaged N-S equation RNG k-120576 turbulence model and structured grid technique are used to numerically analyze the transientflow-field characteristics inside the centrifugal pump volute Based on the quantified parameters of flow field in the volute ofcentrifugal pump the velocity mode contours and oscillation characteristics of the mid-span section of the volute of centrifugalpump are obtained by dynamic mode decomposition (DMD) for the nominal and low flow-rate conditionThe research shows thatthe first-order average flow mode extracted by DMD is the dominant flow structure in the flow field of the volute The second-order and third-order modes are the most important oscillation modes causing unsteady flow in the volute and the characteristicfrequency of the two modes is consistent with the blade passing frequency and the 2x blade passing frequency obtained by the fastFourier transform (FFT) By reconstructing the internal flow field of the volute with the blade passing frequency for the nominalflow-rate condition the periodic variation of the unsteady flow structure in the volute under this frequency is visually reproducedwhich provides some ideas for the study of the unsteady structure in the internal flow field of centrifugal pumps

1 Introduction

The internal flow of the centrifugal pump is a complex three-dimensional unsteady turbulent flow often accompanied byflow separation cavitation hydraulic vibration etc whichaffect the stability of flow in the centrifugal pump Amongthem the rotor-stator interaction will produce periodicpressure fluctuation which will lead to the intensificationof pump vibration noise enhancement and performancedecay seriously affecting the safe and stable operation ofthe unit Especially in the tongue area since the fluid flowfrom the impeller outlet impacts the volute tongue the rotor-stator interaction at this place should be the strongest whichis one of the key areas for the study of unsteady flow incentrifugal pumps For instance Kelder [1] studied the flow inthe volute of a low specific-speed pump both experimentallyand numerically near its design point and found that nearthe design point of this pump the core flow behaves like apotential flow provided that no boundary layer separationoccurs Gonzalez [2] captured the unsteady flow effects insidea centrifugal pump due to the impeller-volute interaction

with a numerical simulation and provided the modelingpossibilities for the prediction of the dynamic interactionbetween the flow at the impeller exit and the volute tongueKeller [3] investigated the unsteady flow in a centrifugalpump with vaneless volute based on PIV measurementsThe obtained data show that the fluid-dynamic blade-tongueinteraction is dominated by high-vorticity sheets (positiveand negative) being shed from the impeller channels espe-cially from the blade trailing edges and their impingementon the tongue tip with subsequent Tan [4] elucidated thedetailed flow field and cavitation effect in the centrifugalpump volute at partial load condition The results showedthat the maximum amplitudes of pressure fluctuations involute at serious cavitation condition are twice that atnoncavitation condition because of the violent disturbancescaused by cavitation shedding and explosion Hamed [5]investigated reducing the radial force the effects of concentricvolute and multivolute geometry on the head efficiency andradial force of a low speed centrifugal pump at off-designconditions The results showed that the triple-volute is themost appropriate volute geometry at off-design conditions

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 2567659 15 pageshttpsdoiorg10115520192567659

2 Mathematical Problems in Engineering

Wang [6] carried out the transient analysis to investigatethe dynamic stress and vibration of volute casing for alarge double-suction centrifugal pump by using the transientfluid-structure interaction theory The results revealed thatfor all operating conditions the maximum stress located atthe volute tongue region whereas the maximum vibrationdisplacement happened close to the shaft hole region and theblade passing frequency and its harmonics were dominant inthe variations of dynamic stress and vibration displacementLiu [7] investigated the pressure fluctuation intensity andvortex characteristic of a mixed-flow pump as turbine atpumpmodewith a tip clearanceHao [8] studied the transientcavitating flows of a mixed-flow PAT (pump as turbine) atpump mode experimentally and numerically Tan [9 10]investigated the role of blade rotational angle in the energyperformance and pressure fluctuation and the influenceof original and T-shape blade end on performance of amixed-flow pump through an experimental measurementand numerical simulation respectively

Dynamic mode decomposition (DMD) is a data-drivenalgorithm for extracting dynamic information from unsteadyexperimental measurements or numerical simulations Forsuch simulations and experiments provide large-scale datait is necessary to understand essential phenomena fromthe data provided So it can be used to analyze the maincharacteristics of complex unsteady flows or to establishlow-order flow-field dynamics models The DMD methodwas first proposed by Schmid [11ndash13] in 2008 and thenthe flow field data obtained by numerical simulation andexperiment were analyzed by DMD method Seena [14]employed the dynamic mode decomposition to analyzethe unsteadiness in extracted modes without the explicitknowledge of evolution operator of the data Hemati [15]presented two algorithms and compared the application ofthe two algorithms on cylinder wake data collected from bothdirect numerical simulations and PIV experiments Tu [16]demonstrated the utility of this approach by presenting novelsampling strategies that increase computational efficiencyand mitigated the effects of noise respectively Pan [17]used a linear combination of a sinusoidal unstable waveand its high-order harmonics as a prototype to take anerror analysis of DMD algorithm The result shows thatthe superimposition of finer structures with less energydominance might damage the estimation accuracy of theprimary structuresrsquo growth rate Diana [18] proposed a newvector-filtering criterion for dynamic modes selection thatis able to extract dynamically relevant flow features fromdynamicmode decomposition of time-resolved experimentalor numerical data Kunihiko [19] described herein someof the dominant techniques for accomplishing these modaldecompositions and analyses that have seen a surge of activityin recent decades and presented a brief overview of several ofthe well-established techniques and clearly lay the frameworkof these methods using familiar linear algebra Liu [20 21]employed the dynamic mode decomposition (DMD) andProper Orthogonal Decomposition (POD) to analyze thecoherent structure of cavitating flow aroundALE 15 hydrofoiland investigated the complicated transient characteristicsof gas-liquid two-phase flow in a multiphase pump under

Table 1 Main design parameters of the centrifugal pump

Design parameter Design valueNominal flow rate 119876d 750m3hRated head119867d 36mRated rotational speed 119899d 1710 rminNumber of blades Z 6Diameter of impeller inlet D1 0245mDiameter of impeller outlet D2 0325mBlade wrap angle 120572 120∘

Blade outlet angle 1205732 27∘

Base circle diameter of volute D3 034m

10 and 20 inlet GVFs by dynamic mode decomposition(DMD)

2 Numerical Model andComputational Method

21 Geometry of the Pump The object of this paper is aconventional single-stage single-suction centrifugal pumpThe main parameters of the pump are listed in Table 1 Thewrap angle 120572 in Table 1 is defined as the angle between theline of the blade inlet edge and the center of the impellerand the line of the blade outlet edge and the center of theimpeller The three-dimensional water body (impeller andvolute) required for the calculation of the model is generatedby three-dimensional design software ProE 50 Accordingto the design parameters the number of blades is 6 and thevolute is a semihelical double volute It should be pointedout that compared with a single volute the presence of thebaffle in the double volute divides the volute into two flowpaths and the volute tongue and the baffle entrance aresymmetrical about the center of rotation High-speed fluidfrom impeller passages still causes unsteady flow in volutewhen it impacts the inlet of baffle Therefore the existenceof baffle is equivalent to the presence of a second tongue indouble volute So this area also needs important attention inthe study of internal flow in volute

22 Comparison of Numerical Simulation and Experiment inthe Centrifugal Pump For the entire computational domain(composed of suction pipe impeller volute and outlet pipe)structured hexahedral mesh is generated using Ansys-ICEMmesh generation tool The overall calculation domain isshown in Figure 1 After the grid independence test the totalnumber of grids was 45 million To calculate the unsteadyflow structures in the centrifugal pump the commercialCFD code Ansys-Fluent is used in the present paper TheRNG k-120576 two-equation model is used under the conditionof satisfying the accuracy The discrete method of the gov-erning equation is the finite volume method The couplingbetween the pressure and velocity is solved using the simplescheme and second-order upwind scheme is used for spatialdiscretization The uniform velocity at the pump inlet is setas the mass flow inlet boundary condition and the outflowat the pump outlet is set as the outlet boundary condition

Mathematical Problems in Engineering 3

VoluteImpeller

Outlet pipe

Suction pipe

Baffle

Tongue

BladeBaffle entrance(the second tongue)

Figure 1 Structured grid of the centrifugal pump

The fluid medium is the normal temperature water and theboundary condition is the no-slip solid wall To guaranteethe accuracy of the result the convergence precision of thecontinuity residual x-velocity residual y-velocity residual z-velocity residual k residual and epsilon residual for steadysimulation is set as 10minus5

As shown in Figure 2 the performance test of thecentrifugal pump was tested on a closed test stand AndFigure 3 shows the headflow-rate relationship and theefficiencyflow-rate relationship curves obtained from bothnumerical simulation and experiment of the centrifugalpump The results show that as the flow rate increases grad-ually the head of the centrifugal pump gradually decreasesand the efficiency first increases and then decreases Forthe different flow-rate conditions at the centrifugal pumpboth the numerically calculated head and efficiency valuesare higher than the experimental values and the errorsfor the nominal flow-rate condition are 58 and 37respectively The errors between numerical simulation andexperiment are due to the fact that the mechanical lossescaused by bearings mechanical seals etc are often ignoredin numerical calculation From the comparison of numericalsimulation and experimental values of head and efficiencyof the centrifugal pump the numerical simulation methodwas used to predict the performance of the centrifugal pumphaving good accuracy of the reference value In addition itcan be seen from Figure 3 that in the efficiencyflow-ratecurve and headflow-rate curve the maximum differenceappears at the nominal flow-rate condition but the valuesobtained by numerical calculation are in good agreementwith the experimental values for the low flow-rate condition

Therefore it can be explained that the centrifugal pump is in astate of off-design condition when it works The main reasonfor this phenomenon is that the design parameters of impellerare unreasonable such as the impeller outlet diameter and theimpeller outlet width being too small

3 Dynamic Mode Decomposition

Dynamicmode decomposition is a method for extracting themode in a flow based on flow-field snapshots so that the flowstructure can be accurately described For the linear flow theDMDmethod can extract themodes that can characterize theglobal flow stability For the nonlinear flow theDMDmethodcan describe the flow structure in which the observations(such as velocity and pressure) dominate The mathematicalderivation process of it is introduced below

Suppose there is a set of observation data matrices thatvary with time

119881N1 = []1 V2 sdot sdot sdot ]119873minus1 V119873] (1)

where N is the total number of snapshots of the flowfield and the column vector vi is the data of the i thsnapshot It is assumed that the snapshot data of the flowfield at two adjacent moments can be represented by a lineartransformation matrix A

]119894+1 = 119860]119894 (2)

As the acquired snapshot data increases we can furtherassume that the vector formedby the snapshot data eventually


Water Tank

Channel Head

Pump

n

M

Outlet Valve

Inlet ValveAdjusting Valve

PI

PI

FMMT

Frequency-inverter motor

Test pump

Inlet pressure gauge

Outlet pressure gaugeOutlet valve

Water tank


Figure 2 Test stand

200 400 600 800 1000

200 400 600 800 1000

100

90

80

70

60

50

40

30

20

10

0

Calculation headExperiment head

Calculation efficiencyExperiment efficiency

Effici

ency

of p

ump

()

Hea

d (m

)

Flow rate (Ｇ3h)

60

55

50

45

40

35

30

25

20

Figure 3 External characteristic curve of the centrifugal pump

tends to be linearizedThus the last flow-field snapshot can berepresented as a linear combination of all previous snapshots

V119873 = 1198861V1 + 1198862V2 + sdot sdot sdot + 119886119873minus1V119873minus1 + 119903 (3)

The matrix form is

V119873 = 119881119873minus11 119886 + 119903 (4)

In formula (4) 119886119879 = [1198861 1198862

sdot sdot sdot 119886119873minus1] r is residual vectorFrom formula (1)1198811198732 = [119860V1 119860V2 sdot sdot sdot 119860V119873minus1] = 119860119881119873minus11 (5)

The new relation can be obtained by combining formula (3)as follows 119860119881119873minus11 = 1198811198732 = 119881119873minus11 119878 + 119903119890119879119873minus1 (6)

In formula (6)

119878 =(((((

0 sdot sdot sdot 0 0 11988611 d 0 d 0 0 119886119873minus3 d 1 0 119886119873minus20 sdot sdot sdot 0 1 119886119873minus1

)))))

(7)

Obviously when the residual vector r is small the eigenvalueof the matrix S is approximated by the eigenvalue of thesystemmatrixATherefore thematrix S is a low-dimensionalapproximation of the systemmatrixA and its eigenvalues can


represent themain eigenvalues of the systemmatrixAMatrixS is usually obtained by QR decomposition of matrix 1198811119873-1if matrix 1198811119873-1 can not guarantee full rank its QR decom-position is not unique and may lead to the irreversibilityof upper triangular matrix R which will ultimately lead tothe impossibility of finding matrix S Perform singular valuedecomposition on matrix 1198811119873-1119881119873minus11 = 119880Σ119881119867 (8)

From formula (5) and formula (7)119880119867119860119880 = 119880119867119860(119880Σ119881119867)119881Σminus1 = 1198801198671198811198732 119881Σminus1 equiv 119878 (9)

From the previous analysis we can see that the eigenval-ues of matrix 119878 can represent the main eigenvalues of systemmatrixA Then the eigenvalues and eigenvectors of matrix 119878are obtained 119878120583119894 = 119908119894120583119894 (10)

Finally the DMDmodes can be obtained as followsΦ119894 = 119880119908119894 (11)

It should be pointed out that the eigenvalue 120583i containsthe information of the mode Φi When the eigenvalues areexpressed in the complex plane the modes on the unit circleof the complex plane are relatively steady while the modeswhose eigenvalues are not on the unit circle are unsteady

The corresponding frequency120596i and growth rate 120593119894 of themodes are defined as follows

120596119894 = Im (ln (120583119894) Δ119905)2120587 (12)

120593119894 = Re( ln (120583119894)Δ119905 ) (13)

According to the similar matrix 119878 obtained by the aboveDMD decomposition the evolution of the flow field can befurther estimated By singular value decomposition (7) ahigh-dimensional system vi can be mapped to a subspace zi119911119894 = 119880119867V119894 (14)

By simple transformation there are119911119894+1 = 119880119867V119894+1 = 119880119867119860V119894 = 119880119867119860119880119911119894 = 119878119911119894 (15)

Let 119908119894be the column vector of matrix W N has the

eigenvalues of the matrix 119878 in its diagonal Then the eigen-decomposition can be expressed as119878 = 119882119873119882minus1 (16)

So from the previous derivation a snapshot at any timeinstant i can be approximated as

V119894 = 119860V119894minus1 = 119880119878119880119867V119894minus1 = 119880119882119873119882minus1119880119867V119894minus1= 119880119882119873119894minus1119882minus1119880119867V1 (17)

From the DMDmode definition formula (10)Φ = 119880119882 (18)

The mode amplitude 120572 is represented as120572 = 119882minus11199111 = 119882minus1119880119867V1 (19)

where 120572i denotes the amplitude of the ith mode whichrepresents the mode contribution to the initial snapshotv1

Substituting formula (17) and (18) into formula (16) theflow field at any time instant can be predicted as

V119894 = 119903sum119895=1

Φ119895 (120583119895)119894minus1 120572119895 (20)

Then its snapshot sequences can be expressed as[V1 V2 sdot sdot sdot V119873minus1] = [Φ1 Φ2 sdot sdot sdot Φ119903]sdot [[[[[[1205721 01205722

d0 120572119903]]]]]][[[[[[[[

1 1205831 sdot sdot sdot 120583119873minus111 1205832 sdot sdot sdot 120583119873minus12 d1 120583119903 sdot sdot sdot 120583119873minus1119903]]]]]]]]

(21)

The dynamic mode decomposition of the velocity field inthe mid-span section of the centrifugal pump volute can beperformed by using the DMD calculation formula derivedabove Firstly we need to edit the formula of DMD calcu-lation in MATLAB and then export the coordinates of gridpoints in the mid-span section and the velocity values at thegrid points in CFD Post From the definition above it can beseen that the set of velocity values at different grid points atthe same time is a column vector of the whole data matrixThe final calculation data matrix is made up of the set ofvelocity values at different times Next the dynamic modedecomposition of the velocitymatrix is then performed usingMATLAB Finally the calculatedmodal values are edited intothe acceptable data format of CFD Post and imported intoCFD Post to make the corresponding velocity contours

4 Results and Discussions

41 Unsteady Pressure Fluctuation Analysis inside the VoluteIn the unsteady calculation the spectrum analysis is per-formed to obtain the characteristic frequency of the unsteadyflow inside the centrifugal pumpThe dimensionless pressurecoefficient Cp is introduced to describe the pressure fluctua-tion characteristics of each monitoring point The expressionis as follows

119862119901 = (119875119894 minus 119875119886V119890)(12) 12058811990622 (22)

where Pi is the static pressure value of the monitoringpoint at a certain time Pa Pave is the average value of staticpressure in one cycle Pa u2 is the circumferential velocity


P4

P1

P2 P3

X

Y

0

28deg

Figure 4 Locations of monitoring points

005

004

003

002

001

0 200 400 600 800 1000 1200 1400 1600

Frequency (Hz)

Ｊ

f1=171Hz

2f1=342Hz

P1

P2

P3

P4

Figure 5 Spectrum characteristic analysis for the low flow-rate condition

at the impeller outlet ms The monitoring points P1 P2 P3and P4 are arranged inside the volute as shown in Figure 4 P1and P3 are set to monitor pressure fluctuation characteristicsof fluid flow in volute tongue and baffle entrance region P2is set to monitor the pressure fluctuation characteristics offluid flow at the volute inlet except for the tongue and thebaffle entrance region and P4 is set to monitor the pressurefluctuation characteristics at the volute outlet By comparingthe pressure fluctuation characteristics at these four pointsthemain factors causing unsteady flow in volute are explored

Figure 5 shows the spectrum analysis results of thepressure fluctuation at the monitoring points inside thecentrifugal pump for the low flow-rate (05119876d) conditionObviously the main frequency of the pressure fluctuationin the centrifugal pump is the blade passing frequency(f 1=171Hz) and the pressure amplitude at 2x blade passingfrequency (2f 1=342HZ) is also prominent Figure 6 showsthe spectrum analysis results of the pressure fluctuation at

the monitoring points inside the centrifugal pump underthe low flow-rate (05119876d) condition For the nominal flow-rate condition the blade passing frequency of the centrifugalpump is f 1=16959Hz and the 2x blade passing frequencyis 2f 1=33917HZ Comparing the results of spectrum anal-ysis for the two operating conditions it can be seen thatthe dominant pressure fluctuation main frequency and thehigh amplitude pressure fluctuation frequency are basicallyconsistent However compared with the nominal flow-ratecondition it produces a certain low frequency fluctuation atless than the blade passing frequency in the low flow-ratecondition

42 Modal Analysis inside the Volute at the Low Flow-RateCondition Based on the previous CFD unsteady calculationresults DMD analysis was performed on 120 flow-field speedsnapshots with time interval Δt=00015s and the DMDresults for the two operating conditions were obtained


005

004

003

002

001

000

0 200 400 600 800 1000 1200 1400 1600

Frequency (Hz)

Ｊ

f1=169 59Hz

2f1=339 17Hz

P1P2

P3P4

Figure 6 Spectrum characteristic analysis for the nominal flow-rate condition

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)

Mode 4

Mode 1

Mode 2Mode 6

Mode 5

Mode 3

04020 06 08minus04minus06minus08 minus02 1minus1Re()

Figure 7 Distribution of DMD eigenvalues for the low flow-ratecondition

Figure 7 shows the distribution of the DMDmode eigenvalue120583i in the complex plane of the unsteady flow field in the mid-span section of the volute of the centrifugal pump at lowflow-rate (05119876d) condition In this figure the real part of theeigenvalue is the horizontal axis and the imaginary part is thevertical axis All the eigenvalues are distributed near the unitcircle and some eigenvalues are distributed on the unit circlewhich means the corresponding modes are neutral stabilityFigure 8 shows the relationship between the correspondingfrequency of the DMDmodes and the correlation coefficientThe correlation coefficient is used to measure the influence ofeach mode on the original flow field In [22] Cj is defined asthe correlation coefficient

119862119895 = Nsum119894=1

100381610038161003816100381610038161003816120572119895 (120583119895)119894minus1100381610038161003816100381610038161003816 10038171003817100381710038171003817Φ119895100381710038171003817100381710038172119865 times Δ119905 (23)

where 120572j is the modal amplitude and 120583j is the eigenvalue andΦ1198952119865 is the mode Frobenius norm After sorting the DMD

modes according to the correlation coefficient the first sixmodes with the highest correlation coefficient can be clearlyseen from Figure 8 Since the corresponding eigenvalue ofthe mode with the largest correlation coefficient is a realnumber its frequency is zero Other eigenvalues are complexnumbers and conjugate pairs appear which can be observedfrom the distribution of eigenvalues Since the parametersdescribing the flow-field information in each mode are theircorresponding real parts modes corresponding to a pair ofconjugate eigenvalues are the same mode The frequenciesof the second and third-order modes are consistent with theblade passing frequency and 2x blade passing frequency ofthe centrifugal pump obtained by the FFT in Figure 5 whichfully demonstrates that the pressure fluctuation frequency inthe original flow field is objectively present and the captureof the characteristic frequency by the DMD is very accurateFrom Figures 7 and 8 it can be clearly seen that the high-energy mode eigenvalues are distributed on the unit circleso the corresponding flow is relatively steady which belongsto the main flow structure in the flow field while the flowcorresponding to the low-energymode eigenvalues which arenot distributed on the unit circle is unsteady and it is not themain flow structure in the flow field

Figure 9 is the first sixth-order mode velocity contourof the mid-span section of the volute for the low flow-rate condition (05119876d) Figure 9(a) is the first-order zero-frequency velocity mode and the eigenvalue correspondingto this mode is real number so the frequency of this modeis zero that is the flow structure is time-average and is theaverage flow mode of the velocity field which shows thedominant flow structure in the original flow field Actuallythe average flow mode can be regarded as the basic structureof the velocity field inside the volute and the original flowfield inside the volute can be regarded as being formed bysuperimposing oscillation modes of different frequencies onthis basic structure It can be seen from the velocity contourthat due to the existence of the volute tongue and the volutebaffle two high-speed fluid regions are distributed at theentrance of the tongue and the baffle which affects thecircumferential distribution of the speed in the volute inletDue to the existence of the baffle the velocity field of the basicflow structure inside the volute is symmetrically distributed


0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

200

400

600

800

1000

1200

1400

1600

Frequency (Hz)

1

23

45 6

f2=171Hzf3=342Hz

0

200

400

600

800

1000

1200

1400

1600

Cor

relat

ion

coeffi

cien

t

Figure 8 Distribution of correlation coefficients for the low flow-rate condition

with respect to the center of rotation which also changes theflow state inside the volute to some extent

Figures 9(b) and 9(c) are velocity contour of the second-order and third-order modes of the mid-span section of thevolute for the low flow-rate condition (05119876d) From thecomparison of correlation coefficients the second- and third-order velocity modes are the two most important modesbesides the first-order zero-frequency modes Therefore thetwo-order velocitymodes are also themain oscillatingmodescausing unsteady flow in the volute From the velocity con-tour in the second- and third-ordermodes it can be seen thatthere are 6 and 12 periodic high and low velocity clusters inthe circumference of the volute inlet respectively Howeverdue to the influence of the tongue and the baffle the highand low velocity fluid regions alternately distributed in thetwo places are weakened to a certain extent The frequenciesof the two-order velocity modes correspond exactly to theblade passing frequency and 2x blade passing frequencyobtained in the FFT which indicates that the DMD capturesthe influence of the centrifugal pump rotor-stator interactionflow structure Figures 9(d) 9(e) and 9(f) are the velocitycontour of the fourth fifth and sixth modes of the mid-spansection of the volute respectively The fourth-order modecaptures the unstable flow structure caused by three highvelocity fluid clusters downstream of the baffle inlet It canbe seen from the fifth order mode velocity contour that highand low velocity fluid clusters alternately distribute at the inletof the tongue and the baffle And high and low velocity fluidclusters alternately distribute in a direction perpendicular tothe line connecting the tongue and the partition in the sixthorder mode velocity contour The distribution indicates thatthere are vortex structures with opposite rotation directionsreflecting the unsteady characteristics of high-order flowinside the volute

43 Modal Analysis inside the Volute at the Nominal Flow-Rate Condition Figures 10 and 11 show the distribution ofthe DMD mode eigenvalues and correlation coefficients of

the unsteady flow field in the mid-span section of the voluteof the centrifugal pump for the nominal flow-rate condition(119876d) In the distribution of eigenvalues it can be seen thatthe eigenvalues of the two modes are in the neutral stablerange of the unit circle but because of their low correlationcoefficients the influence on the original flow field is weak

Figure 12 is the first six-order velocity mode contour ofthe mid-span section of the volute for the nominal flow-rate condition (119876d) where Figure 12(a) is still the first-orderzero-frequency mode which is also the basic structure ofthe velocity field inside the volute The velocity contour ofthe first-order mode is similar to the zero-frequency velocitycontour in the low flow-rate condition A high-speed fluidregion is evenly distributed in the circumferential direction atthe volute inlet and the flow of fluid is still symmetrical aboutthe center of the axis of rotation in the two channels of thevolute The basic flow structure in the volute is still affectedby the presence of the tongue and the baffle but the degreeof influence is greatly reduced compared to the low flow-ratecondition

Figures 12(b) and 12(c) are the second- and third-ordervelocity mode contour of the mid-span section of the volutefor the nominal flow-rate condition (119876d) which are similarto the corresponding modes for the low flow-rate conditionThe twomodes are still the main oscillationmode for causingunsteady flow in the volute Similarly the frequencies of thetwo modes are the same as the blade passing frequency and2x blade passing frequency of the centrifugal pump whichindicates that whether it is in a low or nominal flow-ratecondition the rotor-stator interaction of the impeller and thevolute is the main cause of the unsteady flow in the voluteof the centrifugal pump However unlike the low flow-ratecondition the periodic velocity clusters in the two velocitymodes are almost not affected by the tongue and baffle forthe nominal flow-rate condition

Figures 12(d) 12(e) and 12(f) are the fourth- fifth- andsixth-order velocity mode contour of the mid-span section ofthe volute for the nominal flow-rate condition (119876d) In theseorder velocity mode contours there are 8 10 and 7 periodicvelocity clusters in the inlet of the volute respectively It canbe seen from the velocity contours that the influence range ofthe high and low velocity fluid clusters in these three modeson the flow field is much smaller than that in the second-and third-order modes in the volute which is also consistentwith the distribution of correlation coefficients In additioncompared with similar modes for low flow-rate conditionthe distribution of unsteady fluid clusters is still not affectedby the tongue and baffle in the latter three modes for thenominal flow-rate condition Combining with the secondand third-order modes it shows that the existence of thetongue and baffle can restrain the unsteady flow structureinside the volute when the flow rate of the centrifugal pumpdecreases It should also be pointed out that except thesecond- and third-order modes which cause unsteady flow inthe volute the other three modes are all distributed betweenthe blade passing frequency and 2x blade passing frequencyfor the nominal flow-rate condition while the frequency ofthe fourth-order mode with a high correlation coefficient isless than the blade passing frequency for the low flow-rate


00 01 02 03 04 06 07 08 09 10

DMD Mode 1 of Velocity ms

(a) First-order mode

00 01 02 03 04 06 07 08 09 10


(b) Second-order mode

DMD Mode 3 of Velocity

00 01 02 03 04 06 07 08 09 10

ms

(c) Third-order mode


00 01 02 03 04 06 07 08 09 10

ms

(d) Fourth-order mode


00 01 02 03 04 06 07 08 09 10

ms

(e) Fifth-order mode

00 01 02 03 04 06 07 08 09 10

msDMD Mode 6 of Velocity

(f) Sixth-order mode

Figure 9 DMD velocity mode contours for the low flow-rate condition


Mode 2

Mode 5

Mode 4 Mode 6

Mode 3

Mode 1

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)


Figure 10 Distribution of DMD eigenvalues for the nominal flow-rate condition

0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

160

180

0

20

40

60

80

100

120

140

160

180

Cor

relat

ion

coeffi

cien

t

Frequency (Hz)

1

23

4 56

f2=171Hz f3=342Hz

Figure 11 Distribution of correlation coefficients for the nominal flow-rate condition

condition (05119876d) which indicates that the unsteady flow inthe volute is more prone to occur at low frequencies when theflow rate decreases

44 Reconstruction of Flow Field in Volute Based on DMDIn order to further observe the effect of dynamic modedecomposition on the extraction of flow-field characteristicsin volute of centrifugal pump a reduced order model ofunsteady flow field in volute was established based onformula (19) and the flow field was reconstructed by usingthe obtained dynamic mode decomposition method Theunsteady flow field is reconstructed by the first ten modeswith the highest correlation coefficient obtained by DMDFigures 13 and 14 show the comparison of the reconstructedvelocity contour and the original flow- field velocity contour

at T2 for the low flow-rate condition and the nominal flow-rate condition From the comparison it can be seen that thereconstructed results have a high degree of reduction andidentification for the flow structure in the flow field

Since the first-ordermode is an average flowmode it doesnot change during the entire rotating period of the centrifugalpump In order to further study the unsteady flow structurein a certain mode the mode can be superimposed on thefirst-order average flow mode to observe the oscillation lawof the mode The second-order mode caused by rotor-statorinteraction is superimposed on the average flow mode toreconstruct the flow field of a single mode so as to observethe variation of the unsteady structure with time in themode

Figure 15 shows the reconstructed velocity contour ofthe mode corresponding to the blade passing frequency at


00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10


(d) Fourth order mode

00 01 02 03 04 06 07 08 09 10


(e) Fifth order mode

00 01 02 03 04 06 07 08 09 10


(f) Sixth order mode

Figure 12 DMD velocity mode contours for the nominal flow-rate condition


0 3 5 8 11 13 16 19 21 24

Real flow ms

(a) Original flow field at T2

0 3 5 8 11 13 16 19 21 24

Reconstruction flow ms

(b) Reconstructed flow field at T2

Figure 13 Transient velocity contour inside the volute for the low flow-rate condition

Real flow ms

0 2 4 6 8 10 12 14 16 18



0 2 4 6 8 10 12 14 16 18


Figure 14 Transient velocity contour inside the volute for the nominal flow-rate condition

different times It can be seen from the velocity contourthat six high-speed fluid clusters are uniformly distributedin the circumferential direction of the volute inlet and atthese different times the six high-speed fluid clusters alwaysoccupy a dominant position The motion process of thesix high-speed fluid clusters is consistent with the rotaryscanning process of the blade which indicates that theunsteady flow structure in the volute caused by rotor-stator

interaction is not fixed but is synchronized with the rotationof the blade

5 Conclusions

In this paper the internal flow of the centrifugal pump isnumerically simulated and the velocity field of the mid-span section of the centrifugal pump volute is analyzed by


3T4 T

T4 T2

ms ms

0 2 4 5 7 9 11 12 14 16 0 2 4 5 7 9 11 12 14 16

Reconstruction flow with second mode

ms

0 2 4 5 7 9 11 12 14 16

Reconstruction flow with second mode ms

0 2 4 5 7 9 11 12 14 16



Figure 15 Reconstructed velocity contour with second-order mode for the nominal flow-rate condition

dynamicmode decomposition for the lowflow-rate conditionand the nominal flow-rate condition The characteristicfrequency of the flow field and the flow mode of differentfrequencies are extracted and the reduced order analysis ofthe original flow field is realized In the extracted flowmodesof different frequencies the first-order mode is the flowstructures occupying the dominant position of the originalflow field for the low flow-rate condition and the nominalflow-rate condition The second- and third-order modesare the main oscillation modes of the original flow fieldand the two modesrsquo characteristic frequency is consistent

with the blade passing frequency and 2x blade passingfrequency obtained by FFT which proves that the rotor-statorinteraction between the impeller and the volute is the mainreason for the unsteady flow in the volute Next comparingthe mode velocity contours of each order for the conditionof low flow rate and nominal flow rate when the flow rate ofthe centrifugal pump is reduced the existence of the tongueand the diaphragm will have a certain inhibitory effect onthe unstable flow structure inside the volute but it is moreprone to occur in low frequency unsteady flow Finally theflow field of a single mode is reconstructed by superimposing


the second-order mode to the first-order mode under therated condition And the results show that six high-speedfluid clusters dominated at the volute inlet rotate periodicallywith the blade Generally speaking the combination of DMDmethod and CFD unsteady calculation method is beneficialto understand the flowmechanism in the volute of centrifugalpump by extracting the coherent structure of unsteady flow

Nomenclature119876d Nominal rate flow (m3h)119867d Pump head for nominal rate flow (m)119899d Nominal rotational speed (rmin)t Time step (s)Z Number of bladesD1 Diameter of impeller inlet (m)D2 Diameter of impeller outlet (m)120572 Blade wrap angle1205732 Blade outlet angleD3 Base circle diameter of volute (m)119862119901 Pressure coefficient119875119894 Static pressure value of the monitoring

point at a certain time (Pa)119875119886V119890 Average value of static pressure in onecycle (Pa)1199062 Circumferential velocity at the impelleroutlet (ms)1198811198731 Observation data matrix

N Total number of snapshotsVi Data of the i th snapshotA Linear transformation matrixr Residual vectorS Companion matrixU V Unitary matrixΣ Diagonal matrix120583119894 Eigenvalues of matrix S119908119894 Eigenvectors of matrix S120596119894 Corresponding frequency120593119894 Growth rateBi DMDmodes119911119894 A subspaceW Matrix with 119908119894 as the column vectorN Matrix with 120583119894 as the column vector120572119895 Amplitude of the i th mode119862119895 Correlation coefficientΦ1198952119865 Mode Frobenius norm

Abbreviation

DMD Dynamic mode decompositionFFT Fast Fourier transformationCFD Computational fluid dynamics

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Theauthors have approved the final version of themanuscriptsubmitted There is no financial or personal interest

Acknowledgments

The authors are grateful to the financial support from theNational Natural Science Foundation of China (ResearchProject No 51866009)

References

[1] J D H Kelder R J H Dijkers B P M Van Esch and N PKruyt ldquoExperimental and theoretical study of the flow in thevolute of a low specific-speed pumprdquo Fluid Dynamics Researchvol 28 no 4 pp 267ndash280 2001

[2] J Gonzalez J Fernandez E Blanco and C Santolaria ldquoNumer-ical simulation of the dynamic effects due to impeller-voluteinteraction in a centrifugal pumprdquo Journal of Fluids Engineeringvol 124 no 2 pp 348ndash355 2002

[3] J Keller E Blanco R Barrio and J Parrondo ldquoPIV measure-ments of the unsteady flow structures in a volute centrifugalpump at a high flow raterdquo Experiments in Fluids vol 55 no 10p 1820 2014

[4] L Tan B Zhu Y Wang S Cao and S Gui ldquoNumerical studyon characteristics of unsteady flow in a centrifugal pump voluteat partial load conditionrdquo Engineering Computations (SwanseaWales) vol 32 no 6 pp 1549ndash1566 2015

[5] H Alemi S A Nourbakhsh M Raisee and A F NajafildquoDevelopment of new ldquomultivolute casingrdquo geometries for radialforce reduction in centrifugal pumpsrdquo Engineering Applicationsof Computational Fluid Mechanics vol 9 no 1 pp 1ndash11 2015

[6] F J Wang L X Qu L Y He and J Y Gao ldquoEvaluation offlow-induced dynamic stress and vibration of volute casing fora large-scale double-suction centrifugal pumprdquo MathematicalProblems in Engineering vol 2013 Article ID 764812 9 pages2013

[7] Y Liu and L Tan ldquoTip clearance on pressure fluctuationintensity and vortex characteristic of a mixed flow pump asturbine at pump moderdquo Journal of Renewable Energy vol 129pp 606ndash615 2018

[8] Y Hao and L Tan ldquoSymmetrical and unsymmetrical tip clear-ances on cavitation performance and radial force of a mixedflow pump as turbine at pump moderdquo Journal of RenewableEnergy vol 127 pp 368ndash376 2018

[9] T Lei Y Zhiyi X Yun L Yabin and C Shuliang ldquoRole ofblade rotational angle on energy performance and pressurefluctuation of amixed-flow pumprdquo Proceedings of the Institutionof Mechanical Engineers Part A Journal of Power and Energyvol 231 no 3 pp 227ndash238 2017

[10] T Lei X Zhifeng L Yabin H Yue and X Yun ldquoInfluence ofT-shape tip clearance on performance of a mixed-flow pumprdquoProceedings of the Institution of Mechanical Engineers Part AJournal of Power and Energy vol 232 no 4 pp 386ndash396 2018

[11] P J Schmid and J Sesterhenn ldquoDynamic mode decompositionof numerical and experimental datardquo in Proceedings of the Sixty-First Annual Meeting of the APS Division of Fliuid Mechanics2008

[12] P J Schmid ldquoDynamic mode decomposition of numerical andexperimental datardquo Journal of Fluid Mechanics vol 656 pp 5ndash28 2010


[13] P J Schmid L LiM P Juniper andO Pust ldquoApplications of thedynamic mode decompositionrdquoeoretical and ComputationalFluid Dynamics vol 25 no 1-4 pp 249ndash259 2011

[14] A Seena andH J Sung ldquoDynamicmode decomposition of tur-bulent cavity flows for self-sustained oscillationsrdquo InternationalJournal of Heat and Fluid Flow vol 32 no 6 pp 1098ndash1110 2011

[15] M S Hemati M O Williams and C W Rowley ldquoDynamicmode decomposition for large and streaming datasetsrdquo Physicsof Fluids vol 26 no 11 2014

[16] J H Tu C W Rowley D Luchtenburg S L Brunton andJ N Kutz ldquoOn dynamic mode decomposition theory andapplicationsrdquo Journal of Computational Dynamics vol 1 no 2pp 391ndash421 2014

[17] C Pan D Xue and J Wang ldquoOn the accuracy of dynamicmode decomposition in estimating instability of wave packetrdquoExperiments in Fluids vol 56 no 8 p 164 2015

[18] D A Bistrian and I M Navon ldquoThe method of dynamic modedecomposition in shallow water and a swirling flow problemrdquoInternational Journal for Numerical Methods in Fluids vol 83no 1 pp 73ndash89 2017

[19] K Taira S L Brunton S T Dawson et al ldquoModal analysis offluid flows an overviewrdquo AIAA Journal 2017

[20] M Liu L Tan and S Cao ldquoDynamic mode decomposition ofcavitating flow around ALE 15 hydrofoilrdquo Journal of RenewableEnergy vol 139 pp 214ndash227 2019

[21] M Liu L Tan and S Cao ldquoDynamic mode decomposition ofgas-liquid flow in a rotodynamic multiphase pumprdquo Journal ofRenewable Energy vol 139 pp 1159ndash1175 2019

[22] J Kou andW Zhang ldquoAn improved criterion to select dominantmodes from dynamic mode decompositionrdquo European Journalof Mechanics - BFluids vol 62 pp 109ndash129 2017

Hindawiwwwhindawicom Volume 2018

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International Journal of Mathematics and Mathematical Sciences


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Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

Volume 2018


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Submit your manuscripts atwwwhindawicom


Wang [6] carried out the transient analysis to investigatethe dynamic stress and vibration of volute casing for alarge double-suction centrifugal pump by using the transientfluid-structure interaction theory The results revealed thatfor all operating conditions the maximum stress located atthe volute tongue region whereas the maximum vibrationdisplacement happened close to the shaft hole region and theblade passing frequency and its harmonics were dominant inthe variations of dynamic stress and vibration displacementLiu [7] investigated the pressure fluctuation intensity andvortex characteristic of a mixed-flow pump as turbine atpumpmodewith a tip clearanceHao [8] studied the transientcavitating flows of a mixed-flow PAT (pump as turbine) atpump mode experimentally and numerically Tan [9 10]investigated the role of blade rotational angle in the energyperformance and pressure fluctuation and the influenceof original and T-shape blade end on performance of amixed-flow pump through an experimental measurementand numerical simulation respectively

Dynamic mode decomposition (DMD) is a data-drivenalgorithm for extracting dynamic information from unsteadyexperimental measurements or numerical simulations Forsuch simulations and experiments provide large-scale datait is necessary to understand essential phenomena fromthe data provided So it can be used to analyze the maincharacteristics of complex unsteady flows or to establishlow-order flow-field dynamics models The DMD methodwas first proposed by Schmid [11ndash13] in 2008 and thenthe flow field data obtained by numerical simulation andexperiment were analyzed by DMD method Seena [14]employed the dynamic mode decomposition to analyzethe unsteadiness in extracted modes without the explicitknowledge of evolution operator of the data Hemati [15]presented two algorithms and compared the application ofthe two algorithms on cylinder wake data collected from bothdirect numerical simulations and PIV experiments Tu [16]demonstrated the utility of this approach by presenting novelsampling strategies that increase computational efficiencyand mitigated the effects of noise respectively Pan [17]used a linear combination of a sinusoidal unstable waveand its high-order harmonics as a prototype to take anerror analysis of DMD algorithm The result shows thatthe superimposition of finer structures with less energydominance might damage the estimation accuracy of theprimary structuresrsquo growth rate Diana [18] proposed a newvector-filtering criterion for dynamic modes selection thatis able to extract dynamically relevant flow features fromdynamicmode decomposition of time-resolved experimentalor numerical data Kunihiko [19] described herein someof the dominant techniques for accomplishing these modaldecompositions and analyses that have seen a surge of activityin recent decades and presented a brief overview of several ofthe well-established techniques and clearly lay the frameworkof these methods using familiar linear algebra Liu [20 21]employed the dynamic mode decomposition (DMD) andProper Orthogonal Decomposition (POD) to analyze thecoherent structure of cavitating flow aroundALE 15 hydrofoiland investigated the complicated transient characteristicsof gas-liquid two-phase flow in a multiphase pump under

Table 1 Main design parameters of the centrifugal pump

Design parameter Design valueNominal flow rate 119876d 750m3hRated head119867d 36mRated rotational speed 119899d 1710 rminNumber of blades Z 6Diameter of impeller inlet D1 0245mDiameter of impeller outlet D2 0325mBlade wrap angle 120572 120∘

Blade outlet angle 1205732 27∘

Base circle diameter of volute D3 034m

10 and 20 inlet GVFs by dynamic mode decomposition(DMD)

2 Numerical Model andComputational Method

21 Geometry of the Pump The object of this paper is aconventional single-stage single-suction centrifugal pumpThe main parameters of the pump are listed in Table 1 Thewrap angle 120572 in Table 1 is defined as the angle between theline of the blade inlet edge and the center of the impellerand the line of the blade outlet edge and the center of theimpeller The three-dimensional water body (impeller andvolute) required for the calculation of the model is generatedby three-dimensional design software ProE 50 Accordingto the design parameters the number of blades is 6 and thevolute is a semihelical double volute It should be pointedout that compared with a single volute the presence of thebaffle in the double volute divides the volute into two flowpaths and the volute tongue and the baffle entrance aresymmetrical about the center of rotation High-speed fluidfrom impeller passages still causes unsteady flow in volutewhen it impacts the inlet of baffle Therefore the existenceof baffle is equivalent to the presence of a second tongue indouble volute So this area also needs important attention inthe study of internal flow in volute

22 Comparison of Numerical Simulation and Experiment inthe Centrifugal Pump For the entire computational domain(composed of suction pipe impeller volute and outlet pipe)structured hexahedral mesh is generated using Ansys-ICEMmesh generation tool The overall calculation domain isshown in Figure 1 After the grid independence test the totalnumber of grids was 45 million To calculate the unsteadyflow structures in the centrifugal pump the commercialCFD code Ansys-Fluent is used in the present paper TheRNG k-120576 two-equation model is used under the conditionof satisfying the accuracy The discrete method of the gov-erning equation is the finite volume method The couplingbetween the pressure and velocity is solved using the simplescheme and second-order upwind scheme is used for spatialdiscretization The uniform velocity at the pump inlet is setas the mass flow inlet boundary condition and the outflowat the pump outlet is set as the outlet boundary condition


VoluteImpeller

Outlet pipe

Suction pipe

Baffle

Tongue











]119894+1 = 119860]119894 (2)



Water Tank

Channel Head

Pump

n

M

Outlet Valve


PI

PI

FMMT


Test pump



Water tank


Figure 2 Test stand

200 400 600 800 1000

200 400 600 800 1000

100

90

80

70

60

50

40

30

20

10

0



Effici

ency

of p

ump

()

Hea

d (m

)

Flow rate (Ｇ3h)

60

55

50

45

40

35

30

25

20




The matrix form is

V119873 = 119881119873minus11 119886 + 119903 (4)

In formula (4) 119886119879 = [1198861 1198862



In formula (6)

119878 =(((((


)))))

(7)









120596119894 = Im (ln (120583119894) Δ119905)2120587 (12)

120593119894 = Re( ln (120583119894)Δ119905 ) (13)











V119894 = 119903sum119895=1

Φ119895 (120583119895)119894minus1 120572119895 (20)


d0 120572119903]]]]]][[[[[[[[


(21)




119862119901 = (119875119894 minus 119875119886V119890)(12) 12058811990622 (22)



P4

P1

P2 P3

X

Y

0

28deg


005

004

003

002

001

0 200 400 600 800 1000 1200 1400 1600

Frequency (Hz)

Ｊ

f1=171Hz

2f1=342Hz

P1

P2

P3

P4







005

004

003

002

001

000

0 200 400 600 800 1000 1200 1400 1600

Frequency (Hz)

Ｊ

f1=169 59Hz

2f1=339 17Hz

P1P2

P3P4


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)

Mode 4

Mode 1

Mode 2Mode 6

Mode 5

Mode 3




119862119895 = Nsum119894=1

100381610038161003816100381610038161003816120572119895 (120583119895)119894minus1100381610038161003816100381610038161003816 10038171003817100381710038171003817Φ119895100381710038171003817100381710038172119865 times Δ119905 (23)





0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

200

400

600

800

1000

1200

1400

1600

Frequency (Hz)

1

23

45 6

f2=171Hzf3=342Hz

0

200

400

600

800

1000

1200

1400

1600

Cor

relat

ion

coeffi

cien

t










00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10




00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms


00 01 02 03 04 06 07 08 09 10





Mode 2

Mode 5

Mode 4 Mode 6

Mode 3

Mode 1

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)



0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

160

180

0

20

40

60

80

100

120

140

160

180

Cor

relat

ion

coeffi

cien

t

Frequency (Hz)

1

23

4 56

f2=171Hz f3=342Hz








00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10





0 3 5 8 11 13 16 19 21 24

Real flow ms


0 3 5 8 11 13 16 19 21 24




Real flow ms

0 2 4 6 8 10 12 14 16 18



0 2 4 6 8 10 12 14 16 18





5 Conclusions



3T4 T

T4 T2

ms ms

0 2 4 5 7 9 11 12 14 16 0 2 4 5 7 9 11 12 14 16


ms

0 2 4 5 7 9 11 12 14 16


0 2 4 5 7 9 11 12 14 16











Abbreviation


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









VoluteImpeller

Outlet pipe

Suction pipe

Baffle

Tongue











]119894+1 = 119860]119894 (2)



Water Tank

Channel Head

Pump

n

M

Outlet Valve


PI

PI

FMMT


Test pump



Water tank


Figure 2 Test stand

200 400 600 800 1000

200 400 600 800 1000

100

90

80

70

60

50

40

30

20

10

0



Effici

ency

of p

ump

()

Hea

d (m

)

Flow rate (Ｇ3h)

60

55

50

45

40

35

30

25

20




The matrix form is

V119873 = 119881119873minus11 119886 + 119903 (4)

In formula (4) 119886119879 = [1198861 1198862



In formula (6)

119878 =(((((


)))))

(7)









120596119894 = Im (ln (120583119894) Δ119905)2120587 (12)

120593119894 = Re( ln (120583119894)Δ119905 ) (13)











V119894 = 119903sum119895=1

Φ119895 (120583119895)119894minus1 120572119895 (20)


d0 120572119903]]]]]][[[[[[[[


(21)




119862119901 = (119875119894 minus 119875119886V119890)(12) 12058811990622 (22)



P4

P1

P2 P3

X

Y

0

28deg


005

004

003

002

001

0 200 400 600 800 1000 1200 1400 1600

Frequency (Hz)

Ｊ

f1=171Hz

2f1=342Hz

P1

P2

P3

P4







005

004

003

002

001

000

0 200 400 600 800 1000 1200 1400 1600

Frequency (Hz)

Ｊ

f1=169 59Hz

2f1=339 17Hz

P1P2

P3P4


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)

Mode 4

Mode 1

Mode 2Mode 6

Mode 5

Mode 3




119862119895 = Nsum119894=1

100381610038161003816100381610038161003816120572119895 (120583119895)119894minus1100381610038161003816100381610038161003816 10038171003817100381710038171003817Φ119895100381710038171003817100381710038172119865 times Δ119905 (23)





0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

200

400

600

800

1000

1200

1400

1600

Frequency (Hz)

1

23

45 6

f2=171Hzf3=342Hz

0

200

400

600

800

1000

1200

1400

1600

Cor

relat

ion

coeffi

cien

t










00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10




00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms


00 01 02 03 04 06 07 08 09 10





Mode 2

Mode 5

Mode 4 Mode 6

Mode 3

Mode 1

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)



0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

160

180

0

20

40

60

80

100

120

140

160

180

Cor

relat

ion

coeffi

cien

t

Frequency (Hz)

1

23

4 56

f2=171Hz f3=342Hz








00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10





0 3 5 8 11 13 16 19 21 24

Real flow ms


0 3 5 8 11 13 16 19 21 24




Real flow ms

0 2 4 6 8 10 12 14 16 18



0 2 4 6 8 10 12 14 16 18





5 Conclusions



3T4 T

T4 T2

ms ms

0 2 4 5 7 9 11 12 14 16 0 2 4 5 7 9 11 12 14 16


ms

0 2 4 5 7 9 11 12 14 16


0 2 4 5 7 9 11 12 14 16











Abbreviation


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









Water Tank

Channel Head

Pump

n

M

Outlet Valve


PI

PI

FMMT


Test pump



Water tank


Figure 2 Test stand

200 400 600 800 1000

200 400 600 800 1000

100

90

80

70

60

50

40

30

20

10

0



Effici

ency

of p

ump

()

Hea

d (m

)

Flow rate (Ｇ3h)

60

55

50

45

40

35

30

25

20




The matrix form is

V119873 = 119881119873minus11 119886 + 119903 (4)

In formula (4) 119886119879 = [1198861 1198862



In formula (6)

119878 =(((((


)))))

(7)









120596119894 = Im (ln (120583119894) Δ119905)2120587 (12)

120593119894 = Re( ln (120583119894)Δ119905 ) (13)











V119894 = 119903sum119895=1

Φ119895 (120583119895)119894minus1 120572119895 (20)


d0 120572119903]]]]]][[[[[[[[


(21)




119862119901 = (119875119894 minus 119875119886V119890)(12) 12058811990622 (22)



P4

P1

P2 P3

X

Y

0

28deg


005

004

003

002

001

0 200 400 600 800 1000 1200 1400 1600

Frequency (Hz)

Ｊ

f1=171Hz

2f1=342Hz

P1

P2

P3

P4







005

004

003

002

001

000

0 200 400 600 800 1000 1200 1400 1600

Frequency (Hz)

Ｊ

f1=169 59Hz

2f1=339 17Hz

P1P2

P3P4


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)

Mode 4

Mode 1

Mode 2Mode 6

Mode 5

Mode 3




119862119895 = Nsum119894=1

100381610038161003816100381610038161003816120572119895 (120583119895)119894minus1100381610038161003816100381610038161003816 10038171003817100381710038171003817Φ119895100381710038171003817100381710038172119865 times Δ119905 (23)





0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

200

400

600

800

1000

1200

1400

1600

Frequency (Hz)

1

23

45 6

f2=171Hzf3=342Hz

0

200

400

600

800

1000

1200

1400

1600

Cor

relat

ion

coeffi

cien

t










00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10




00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms


00 01 02 03 04 06 07 08 09 10





Mode 2

Mode 5

Mode 4 Mode 6

Mode 3

Mode 1

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)



0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

160

180

0

20

40

60

80

100

120

140

160

180

Cor

relat

ion

coeffi

cien

t

Frequency (Hz)

1

23

4 56

f2=171Hz f3=342Hz








00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10





0 3 5 8 11 13 16 19 21 24

Real flow ms


0 3 5 8 11 13 16 19 21 24




Real flow ms

0 2 4 6 8 10 12 14 16 18



0 2 4 6 8 10 12 14 16 18





5 Conclusions



3T4 T

T4 T2

ms ms

0 2 4 5 7 9 11 12 14 16 0 2 4 5 7 9 11 12 14 16


ms

0 2 4 5 7 9 11 12 14 16


0 2 4 5 7 9 11 12 14 16











Abbreviation


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018















120596119894 = Im (ln (120583119894) Δ119905)2120587 (12)

120593119894 = Re( ln (120583119894)Δ119905 ) (13)











V119894 = 119903sum119895=1

Φ119895 (120583119895)119894minus1 120572119895 (20)


d0 120572119903]]]]]][[[[[[[[


(21)




119862119901 = (119875119894 minus 119875119886V119890)(12) 12058811990622 (22)



P4

P1

P2 P3

X

Y

0

28deg


005

004

003

002

001

0 200 400 600 800 1000 1200 1400 1600

Frequency (Hz)

Ｊ

f1=171Hz

2f1=342Hz

P1

P2

P3

P4







005

004

003

002

001

000

0 200 400 600 800 1000 1200 1400 1600

Frequency (Hz)

Ｊ

f1=169 59Hz

2f1=339 17Hz

P1P2

P3P4


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)

Mode 4

Mode 1

Mode 2Mode 6

Mode 5

Mode 3




119862119895 = Nsum119894=1

100381610038161003816100381610038161003816120572119895 (120583119895)119894minus1100381610038161003816100381610038161003816 10038171003817100381710038171003817Φ119895100381710038171003817100381710038172119865 times Δ119905 (23)





0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

200

400

600

800

1000

1200

1400

1600

Frequency (Hz)

1

23

45 6

f2=171Hzf3=342Hz

0

200

400

600

800

1000

1200

1400

1600

Cor

relat

ion

coeffi

cien

t










00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10




00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms


00 01 02 03 04 06 07 08 09 10





Mode 2

Mode 5

Mode 4 Mode 6

Mode 3

Mode 1

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)



0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

20

40

60

80

100

120

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160

180

0

20

40

60

80

100

120

140

160

180

Cor

relat

ion

coeffi

cien

t

Frequency (Hz)

1

23

4 56

f2=171Hz f3=342Hz








00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10





0 3 5 8 11 13 16 19 21 24

Real flow ms


0 3 5 8 11 13 16 19 21 24




Real flow ms

0 2 4 6 8 10 12 14 16 18



0 2 4 6 8 10 12 14 16 18





5 Conclusions



3T4 T

T4 T2

ms ms

0 2 4 5 7 9 11 12 14 16 0 2 4 5 7 9 11 12 14 16


ms

0 2 4 5 7 9 11 12 14 16


0 2 4 5 7 9 11 12 14 16











Abbreviation


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









P4

P1

P2 P3

X

Y

0

28deg


005

004

003

002

001

0 200 400 600 800 1000 1200 1400 1600

Frequency (Hz)

Ｊ

f1=171Hz

2f1=342Hz

P1

P2

P3

P4







005

004

003

002

001

000

0 200 400 600 800 1000 1200 1400 1600

Frequency (Hz)

Ｊ

f1=169 59Hz

2f1=339 17Hz

P1P2

P3P4


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)

Mode 4

Mode 1

Mode 2Mode 6

Mode 5

Mode 3




119862119895 = Nsum119894=1

100381610038161003816100381610038161003816120572119895 (120583119895)119894minus1100381610038161003816100381610038161003816 10038171003817100381710038171003817Φ119895100381710038171003817100381710038172119865 times Δ119905 (23)





0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

200

400

600

800

1000

1200

1400

1600

Frequency (Hz)

1

23

45 6

f2=171Hzf3=342Hz

0

200

400

600

800

1000

1200

1400

1600

Cor

relat

ion

coeffi

cien

t










00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10




00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms


00 01 02 03 04 06 07 08 09 10





Mode 2

Mode 5

Mode 4 Mode 6

Mode 3

Mode 1

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)



0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

20

40

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160

180

0

20

40

60

80

100

120

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160

180

Cor

relat

ion

coeffi

cien

t

Frequency (Hz)

1

23

4 56

f2=171Hz f3=342Hz








00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10





0 3 5 8 11 13 16 19 21 24

Real flow ms


0 3 5 8 11 13 16 19 21 24




Real flow ms

0 2 4 6 8 10 12 14 16 18



0 2 4 6 8 10 12 14 16 18





5 Conclusions



3T4 T

T4 T2

ms ms

0 2 4 5 7 9 11 12 14 16 0 2 4 5 7 9 11 12 14 16


ms

0 2 4 5 7 9 11 12 14 16


0 2 4 5 7 9 11 12 14 16











Abbreviation


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









005

004

003

002

001

000

0 200 400 600 800 1000 1200 1400 1600

Frequency (Hz)

Ｊ

f1=169 59Hz

2f1=339 17Hz

P1P2

P3P4


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)

Mode 4

Mode 1

Mode 2Mode 6

Mode 5

Mode 3




119862119895 = Nsum119894=1

100381610038161003816100381610038161003816120572119895 (120583119895)119894minus1100381610038161003816100381610038161003816 10038171003817100381710038171003817Φ119895100381710038171003817100381710038172119865 times Δ119905 (23)





0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

200

400

600

800

1000

1200

1400

1600

Frequency (Hz)

1

23

45 6

f2=171Hzf3=342Hz

0

200

400

600

800

1000

1200

1400

1600

Cor

relat

ion

coeffi

cien

t










00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10




00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms


00 01 02 03 04 06 07 08 09 10





Mode 2

Mode 5

Mode 4 Mode 6

Mode 3

Mode 1

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)



0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

160

180

0

20

40

60

80

100

120

140

160

180

Cor

relat

ion

coeffi

cien

t

Frequency (Hz)

1

23

4 56

f2=171Hz f3=342Hz








00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10





0 3 5 8 11 13 16 19 21 24

Real flow ms


0 3 5 8 11 13 16 19 21 24




Real flow ms

0 2 4 6 8 10 12 14 16 18



0 2 4 6 8 10 12 14 16 18





5 Conclusions



3T4 T

T4 T2

ms ms

0 2 4 5 7 9 11 12 14 16 0 2 4 5 7 9 11 12 14 16


ms

0 2 4 5 7 9 11 12 14 16


0 2 4 5 7 9 11 12 14 16











Abbreviation


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

200

400

600

800

1000

1200

1400

1600

Frequency (Hz)

1

23

45 6

f2=171Hzf3=342Hz

0

200

400

600

800

1000

1200

1400

1600

Cor

relat

ion

coeffi

cien

t










00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10




00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms


00 01 02 03 04 06 07 08 09 10





Mode 2

Mode 5

Mode 4 Mode 6

Mode 3

Mode 1

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)



0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

20

40

60

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100

120

140

160

180

0

20

40

60

80

100

120

140

160

180

Cor

relat

ion

coeffi

cien

t

Frequency (Hz)

1

23

4 56

f2=171Hz f3=342Hz








00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10





0 3 5 8 11 13 16 19 21 24

Real flow ms


0 3 5 8 11 13 16 19 21 24




Real flow ms

0 2 4 6 8 10 12 14 16 18



0 2 4 6 8 10 12 14 16 18





5 Conclusions



3T4 T

T4 T2

ms ms

0 2 4 5 7 9 11 12 14 16 0 2 4 5 7 9 11 12 14 16


ms

0 2 4 5 7 9 11 12 14 16


0 2 4 5 7 9 11 12 14 16











Abbreviation


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10




00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms



00 01 02 03 04 06 07 08 09 10

ms


00 01 02 03 04 06 07 08 09 10





Mode 2

Mode 5

Mode 4 Mode 6

Mode 3

Mode 1

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)



0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

160

180

0

20

40

60

80

100

120

140

160

180

Cor

relat

ion

coeffi

cien

t

Frequency (Hz)

1

23

4 56

f2=171Hz f3=342Hz








00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10





0 3 5 8 11 13 16 19 21 24

Real flow ms


0 3 5 8 11 13 16 19 21 24




Real flow ms

0 2 4 6 8 10 12 14 16 18



0 2 4 6 8 10 12 14 16 18





5 Conclusions



3T4 T

T4 T2

ms ms

0 2 4 5 7 9 11 12 14 16 0 2 4 5 7 9 11 12 14 16


ms

0 2 4 5 7 9 11 12 14 16


0 2 4 5 7 9 11 12 14 16











Abbreviation


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









Mode 2

Mode 5

Mode 4 Mode 6

Mode 3

Mode 1

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Im(

)



0 50 100 150 200 250 300

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

160

180

0

20

40

60

80

100

120

140

160

180

Cor

relat

ion

coeffi

cien

t

Frequency (Hz)

1

23

4 56

f2=171Hz f3=342Hz








00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10





0 3 5 8 11 13 16 19 21 24

Real flow ms


0 3 5 8 11 13 16 19 21 24




Real flow ms

0 2 4 6 8 10 12 14 16 18



0 2 4 6 8 10 12 14 16 18





5 Conclusions



3T4 T

T4 T2

ms ms

0 2 4 5 7 9 11 12 14 16 0 2 4 5 7 9 11 12 14 16


ms

0 2 4 5 7 9 11 12 14 16


0 2 4 5 7 9 11 12 14 16











Abbreviation


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10



00 01 02 03 04 06 07 08 09 10





0 3 5 8 11 13 16 19 21 24

Real flow ms


0 3 5 8 11 13 16 19 21 24




Real flow ms

0 2 4 6 8 10 12 14 16 18



0 2 4 6 8 10 12 14 16 18





5 Conclusions



3T4 T

T4 T2

ms ms

0 2 4 5 7 9 11 12 14 16 0 2 4 5 7 9 11 12 14 16


ms

0 2 4 5 7 9 11 12 14 16


0 2 4 5 7 9 11 12 14 16











Abbreviation


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









0 3 5 8 11 13 16 19 21 24

Real flow ms


0 3 5 8 11 13 16 19 21 24




Real flow ms

0 2 4 6 8 10 12 14 16 18



0 2 4 6 8 10 12 14 16 18





5 Conclusions



3T4 T

T4 T2

ms ms

0 2 4 5 7 9 11 12 14 16 0 2 4 5 7 9 11 12 14 16


ms

0 2 4 5 7 9 11 12 14 16


0 2 4 5 7 9 11 12 14 16











Abbreviation


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









3T4 T

T4 T2

ms ms

0 2 4 5 7 9 11 12 14 16 0 2 4 5 7 9 11 12 14 16


ms

0 2 4 5 7 9 11 12 14 16


0 2 4 5 7 9 11 12 14 16











Abbreviation


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018













Abbreviation


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018


























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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Study on Flow Characteristics in Volute of …downloads.hindawi.com/journals/mpe/2019/2567659.pdfcentrifugal pump for the low ow-rate (. d) condition. Obviously, the main frequency