Journal of Fluids and Structures 35 (2012) 89–104 Contents lists available at SciVerse ScienceDirect Journal of Fluids and Structures journal homepage: www.elsevier.com/locate/jfs Investigation of unsteady flow-induced impeller oscillations of a single-blade pump under off-design conditions J. Pei a,b, H.J. Dohmen b, S.Q. Yuan a, F.-K. Benra b,n a b Research Center of Fluid Machinery Engineering and Technology, Jiangsu University, Zhenjiang 212013, China Department of Mechanical Engineering, Faculty of Engineering, University of Duisburg-Essen, Duisburg 47048, Germany a r t i c l e i n f o abstract Article history: Received 21 September 2011 Accepted 8 August 2012 Available online 30 August 2012 The periodically unsteady flow-induced impeller oscillations for a single-blade pump are investigated both numerically and experimentally under off-design conditions. A partitioned strategy with load transfer method is selected for achieving successful fluid– structure interaction (FSI) simulations with strong two-way coupling. Three-dimensional, unsteady Reynolds-averaged Navier–Stokes equations are solved with a shear stress transport turbulence model for the fluid side, while a transient structure dynamic analysis with the finite element method is employed for the structure side. Radial deflections of the pump impeller are successfully measured using proximity sensors to validate the FSI results. The comparison of the deflection results focuses on both phase and amplitude aspects under different operational conditions. The FSI calculation results are confirmed by the experiment, but deviations are still observed for about half of an impeller rotation. Therefore, a rigorous analysis of the comparison between the angles of the obtained x and y components is carried out to understand the cause of the deviation. Meanwhile, the transient pressure measured at the casing by both computational fluid dynamics and experimental methods is qualitatively analyzed. Furthermore, hydrodynamic forces are also analyzed considering a strong FSI effect in both the rotating and stationary coordinate frame under off-design conditions to understand the behavior of the transient excitation forces, which directly lead to the rotor deflection. & 2012 Elsevier Ltd. All rights reserved. Keywords: Single-blade pump impeller Fluid–structure interaction Coupled solution Hydrodynamic forces Fluid dynamics Structural analysis 1. Introduction A centrifugal pump is one of the most important energy conversion devices and is widely used in almost all industrial and agricultural applications. Single-blade pumps with a special impeller design are used for transporting water containing large amounts of solids and fibers, and are called no-clogging sewage water pumps. This type of impeller geometry, however, results in a strong asymmetrical unsteady flow and consistent periodically unsteady excitation forces under design condition and an even stronger asymmetrical unsteady flow under off-design conditions, due to a hydrodynamic unbalance (Agostinelli et al., 1960; Aoki, 1984; Okamura, 1980). This leads to impeller oscillations and alternating stresses, designated as a flow-induced vibration problem (Guelich, 2007). Not only the impeller itself, but also other pump components, such as the casing and bearings, may be damaged by these oscillations and the corresponding transmitted vibrations. n Corresponding author. Tel.: þ49 203 379 3030. E-mail address: friedrich.benra@uni-due.de (F.-K. Benra). 0889-9746/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfluidstructs.2012.08.005 90 J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 Nomenclature a b2 ds Fref Fx, Fy Fx, Fc H Hs Hdes n ndes pc Q Qdes r2 S Sx, Sy load component blade height at impeller outlet pipe diameter at pump suction reference force force components along x and y axes force components along x and c axes delivery head supply head delivery head under design condition rotation speed nominal rotation speed pressure at casing flow rate flow rate under design condition radius at impeller outlet rotor deflection rotor deflection components along x and y axes Sx,0, Sy,0 initial values of rotor deflection components along x and y axes Sx,test, Sy,test test oscillation values of rotor deflection components along x and y axes Sx,wet, Sy,wet values of rotor deflection components along x and y axes under ‘‘wet’’ condition Sx,dry, Sy,dry values of rotor deflection components along x and y axes under ‘‘dry’’ condition u2 circumference velocity at impeller outlet x, y coordinates in stationary frame Greek letters D x, c r f j difference coordinates in rotation frame density convergence criterion angle of rotation To solve this problem, the analysis of the unsteady flow field and structural dynamic response of a pump impeller should consider the fluid–structure interaction (FSI), which in this case, refers to the interaction between the complex inner flow and structures of centrifugal pumps. In FSI, the hydraulic excitation changes the kinetic characteristics of the structures and leads to the deformation of these structures, which can affect the distribution of the pump inner flow field. Although research on the application of computational fluid dynamics (CFD) and computational structural dynamics (CSD) is yet in its nascent stages, the maturity of these methods is sufficient to enable the numerical simulation of FSI. To solve complicated FSI problems, two strategies can be used depending on the physical nature of the interaction (Felippa and Park, 1980; Piperno and Farhat, 2001): the monolithic approach and partitioned approach. The monolithic method requires a straightforward solution for all unknowns of the overall coupled system and solves the resulting system of equations with a complete tangent stiffness matrix. In this way, all interaction effects between the dependent equations are addressed in one solver. This approach is ideal when the physical interactions are strongly nonlinear; however, it is currently difficult to implement because of some severe drawbacks, such as the complex modeling required for both fluid and structure fields and the large computational resources required. In the partitioned method, the equations governing the flow and the displacement of the structure are solved separately in different solvers without any limitation. This enables us to exploit the advantages of mature solvers for both CFD and CSD, which have been developed in recent years for engineering applications, precluding the need for developing a specialized solver for an FSI problem. The interaction effects between both physical fields are represented by an exchange of loads (total force and mesh displacement) at the common interface. For the solid component, the natural view is the material (Lagrangian) description, and for the fluid, it is the spatial (Eulerian) description. Combining these views requires a mixed description for the partitioned method, and for this purpose, the arbitrary Lagrangian–Eulerian description (Belytschko and Kennedy, 1978; Belytschko et al., 1980; Hughes et al., 1981) is usually selected. Although FSI problems have received significant attention, only a small number of research studies have been conducted to experimentally validate complex FSI problems in engineering fields because in some cases, the deformations are negligible or cannot be easily measured for moving or rotating structures due to the limitations of the measurement methods and equipment. In the research area of fluid machinery, some numerical research was conducted with a coupled method, but only a few studies considered the vibration problems of pump machinery in which water was the operational medium, such as the works by Benra (2006), Benra et al. (2006), Benra and Dohmen (2007), Campbell and Paterson (2011), Gnesin and Rzadkowski (2002), Kato et al. (2005), Langthjem and Olhoff (2004), and Muench et al. (2010). These numerical and experimental studies offer insight on flow-induced vibration and noise phenomenon in fluid machinery; however, only a few considered the integrated interaction between the fluid and solid with a strongly coupled analysis. Furthermore, for off-design operational conditions, no sufficient numerical and experimental results were compared and analyzed in detail. The present research is an improvement over former works, with more thorough analyses and the following new aspects: (1) A full two-way coupling calculation, which includes stagger iterations between the two solvers for each time step, was conducted to obtain strongly coupled results for an impeller deflection calculation. The stagger iterations were not J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 91 stopped until the load transfer components converged. The selection of the Rayleigh and numerical damping for the finite element method (FEM) solver was evaluated to obtain stable and reasonably strong FSI results. (2) A straight pipe at the pump suction, instead of an elbow, was used to eliminate the influence of the asymmetrical inflow highlighted in the previous experimental works. Moreover, the sampling frequency of the impeller deflection signal acquisition was increased to capture as many features of the impeller oscillation signal as possible. To eliminate uncertainty and error, a specific phase-averaged program was developed using LabVIEW. A new type of Butterworth low-pass filter without a phase shift was developed to obtain results without phase distortion. Furthermore, a highly accurate transient pressure sensor was used to measure the transient pressure in the casing to validate the timedependent fluid flow calculation results only in the CFD solver. (3) This study focused on the two-way coupling results under off-design conditions, and a comparison between the coupled and measured results at low rotation speeds is presented for the first time. Not only the amplitude but also the phase of the impeller oscillation is discussed in detail. In addition, the coupled and measured deflection results were analyzed at different angles to determine the reasons for the deviations. The numerical and experimental transient pressure results were analyzed, and the x and y component curves of the impeller deflection results were thoroughly investigated. Relationships between the FSI and CFD results were clarified, and additional possible reasons for the deviation were discussed. 2. Coupling simulation and experimental validation 2.1. Coupled simulation strategy In this study, the partitioned approach, in which a two-way load transfer coupling method is applied for the staggered solution procedure, is selected for the strongly coupled solution, because the two domains are separated by an apparent boundary where the interaction between the fluid and structure is effected. The ANSYS Multi-field solver system 12.1 including the mechanical and CFX solvers is used. In Fig. 1, the scheme of the partitioned FSI simulation system is depicted, which includes three parts: the time domain of the coupled solution, stagger iteration loops, and field solution loops. A time domain that includes a certain number of time steps is used for defining the time duration of physical problems. Within every time step, which corresponds to a certain impeller position, stagger iteration loops, each of which includes a certain number of field solution loops, are processed. Within each stagger iteration loop, structural dynamics simulation of the rotor using the FEM and unsteady flow field calculations using the finite volume method are performed alternately, along with data transfers between two fields. In the present case, only the pressure force is included in the fluid load. The structural deformation influences the flow by changing the flow geometry, and the influence of vibration velocity on the moment is not considered. With the consideration of ‘‘critical speed’’, modal analysis is very necessary as the beginning of the dynamic analysis. The first natural frequency of the rotor was 127.78 Hz. Comparing this natural frequency with test rotating speed frequency value 24 Hz, the latter frequency is much smaller than the earlier one. Therefore, no resonance would happen in this case. In addition, the dominant hydrodynamic force component changed slowly in comparison to the first rotor natural frequency, and the structure’s response may be determined with static analysis. This assumption can be clearly confirmed by the comparison between the calculated rotor deflection results with static and dynamic analysis for each time step in Fig. 2, and the structure’s response can be assumed to vary slowly with respect to time. Therefore, the assumption mentioned above that only the deflection displacement was considered in the coupled calculations can be acceptable. Fig. 1. Partitioned method scheme for FSI simulation. 92 J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 Fig. 2. Comparison of rotor deflection results with static and dynamic analysis for Q¼ 22 l/s, n ¼1440 rpm. The stagger iteration loop will not stop until the loads transferred across the physical interface converge. The convergence criterion for the load transfer procedure is defined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P ðanew aold Þ2 pP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f¼ , ð1Þ a2new where aold is the load component transferred in the last stagger iteration, anew is the load component transferred in the current stagger iteration, and the summation is carried out over all the individual load component values transferred from different points in space. Each quantity is considered to converge when f is less than fmin , where fmin is the convergence target. The convergence of each quantity transferred across the interface is denoted as e, which is defined by e¼ logðf=fmin Þ , logð10=fmin Þ where 0 o fmin o 1: ð2Þ This implies that each quantity has converged when the reported convergence reaches a negative value. 2.2. Coupled simulation setup The test pump and design parameters are shown in Fig. 3 and Table 1, respectively. For the fluid calculation, threedimensional, unsteady Reynolds-averaged Navier–Stokes equations were solved by the CFD code CFX 12.1. The structured grids, shown in Fig. 4, for the computational domains were generated using the grid generation tool ICEM-CFD 12.1. The impeller side chambers were also included in the grid to take the leakage flow effect into account, as shown in Fig. 5. The total pressure under the stationary frame and the flow direction were set at the inlet, and the mass flow rate was given at the outlet. The turbulence was simulated by the shear stress transport turbulence model. The turbulence intensity was set equal to 5%. The interface between the impeller and the casing was set to ‘‘transient rotor–stator’’ to capture the transient rotor–stator interaction in the flow, because the relative position between the impeller and the casing changes with each time step for this kind of an interface. A smooth wall condition was used as the wall function. The time step for the transient simulation was selected as 3.47225 10 4 s for a nominal rotation speed, n ¼1440 rpm, while 4 10 4 s and 5 10 4 s were selected for the rotation speeds of 1250 rpm and 1000 rpm, respectively, corresponding to an angular rotation of Dj ¼31. Within each time step, the iteration stops when the maximum residual was less than 10 3. Transient simulations for eight impeller rotations were conducted and used as initial conditions for the coupled simulation. To obtain the periodic stable rotor deflection results, two-way coupling calculations were performed for six impeller rotations, and a typical result is shown in Fig. 6. Periodically stable results were achieved, and therefore, the calculation results of the sixth impeller rotation were recorded for later analysis. The impeller structure mesh for FEM analysis, which contains 54 034 nodes, was generated by the ANSYS Structure Mesher, and a hex dominant grid with elements of Solid186 was selected for the load transfer coupled simulation, as shown in Fig. 7. The fluid–structure interfaces for all wetted surfaces of the blade, shroud, and hub, shown in yellow, were defined, including the side chamber wetted surfaces of the shroud and hub. Furthermore, the face fixed support boundary conditions were indicated in blue in Fig. 7; therefore, the bearings mounted here were treated as a rigid structure. Additional transient structural computation information is listed in Table 2. The Hilber–Hughes–Taylor method (HHT) was J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 93 Fig. 3. Test pump. Table 1 Pump design parameters. Head Flow rate Rotation speed Pipe diameter Hdes Qdes ndes ds 8m 33.3 l/s 1440 rpm 100 mm Fig. 4. Computational grid view of test pump. employed for time integration of the transient dynamic equilibrium equation, which is an improved algorithm of the Newmark time integration method. In addition, Rayleigh damping was used in the solving system to apply the damping effect to the transient analysis. Two damping values, Alpha and Beta, are included in Rayleigh damping. Alpha is a massrelated Rayleigh damping coefficient, which should be considered at joints and can be neglected for this case, whereas Beta is a stiffness-related coefficient. Beta damping cannot be determined directly by any theory but by experience or the trialand-error method. Numerical damping is associated with the time-stepping schemes used for integrating second-order systems of equations over time. According to the physical problem, the lowest possible value of numerical damping that dampens nonphysical responses without significantly affecting the final solution should be used (Table 2). Finally, for the external load transfer parameters, the under relaxation factor was fixed to 0.75 for all data transferred between the two solvers to control the solving convergence process, and the convergence target for the load transfer was 0.01. In addition, the non-matching area fractions between the fluid side and the structure side during the interpolation mapping process for the blade, hub, and shroud interfaces were less than 0.8%, indicating that the two sides of the fluid– structure interface for the load transfer had almost the same shape, and less error will be produced in the load transfer. Meanwhile, for each interface, different data transfer methods of mesh interpolation were selected for different types of variables, according to the behavior of the variables, to obtain reasonable results. Profile preserving, which maps the profile of a variable from one mesh to the other as accurately as possible, was used for the displacement variable 94 J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 Fig. 5. Cross section of computational domain. Fig. 6. Typical convergence history of a two way coupling simulation. Fig. 7. Impeller structure and FEM grid. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.) transferred from the CSD solver to the CFD solver. A conservation method, which ensures that a total quantity passing across the interface is conserved, was selected for the force variable transferred in the opposite direction. Furthermore, the CFX solver was started before the mechanical solver at the beginning of the stagger iteration. J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 95 Table 2 Structural computation information. Time integration method Beta damping value Numerical damping Material density Material elastic modulus Material Poisson ratio HHT method 0.1% 0.4 7850 kg/m3 200 GPa 0.3 Fig. 8. Sketch of test rig. 2.3. Test rig and instruments To validate the coupling simulation results of the impeller deflections, a commercial centrifugal pump was used. The test single-blade pump was a horizontal-type commercial water pump. The sketch of the complete test rig is shown in Fig. 8. The pump was arranged in an open loop, and the supply head Hs was approximately 1.5 m. The tank contains about 8 m3 of pure industrial water. An electromagnetic flow meter was used for flow rate measurement and was mounted on the inlet pipeline of the pump, and the operational conditions were adjusted by a control valve located on the pump outlet pipeline. An adjustable speed motor of 4.4 kW was used to drive the pump up to a maximal speed of 1440 rpm. Both the pump and motor were mounted on the metal foundation support. Several sensors were utilized on the test rig for different measurements. Static pressure sensors were mounted on the pipe to measure the static pressure at the suction and pressure sides, and the pump head were calculated by these measured values. An absolute static pressure sensor with a range of 0–1.6 bar was used for the suction side of the pump, and a relative pressure sensor with a range of 0–4 bar was used for the pressure side. In addition, a Kulite XCL-093 absolute transient pressure sensor with a range of 0–3.5 bar was mounted on the pump casing to monitor the pressure fluctuation inside the pump casing. The sensor is based on a fully active four-arm Wheatstone bridge, whose full-scale output is 100 mV, output sensitivity is 10 mV/V, and excitation voltage is 10 VDC/AC. An amplifier was used to amplify the signal from the sensor, and the output voltage range was 0–10 V. To ensure precise measurement, the transient pressure sensor was calibrated by a high-accuracy pressure calibrator. To increase the accuracy of the system, the absolute pressure range of 1–2.75 bar, which was the possible pressure range in this case, was amplified to the output voltage range of 0–10 V. The position of the mounted pressure sensor is shown in Fig. 9, and the calibration points and the linear trend line are shown in Fig. 10. A good linear behavior between the output voltage and standard input pressure value of the sensor can be observed. In addition, an inductive pulse sensor was used to determine the phasing of the impeller. To measure the radial deflection of the impeller, two non-contacting displacement measuring chains were used for the two perpendicular directions in the stationary coordinate frame, identified as the x and y direction, and each chain included a displacement sensor, extension cables, and an oscillator. The test principle is based on electromagnetic induction and reaction. The positions of the sensors mounted on the pump and the defined coordinate frames are shown in Fig. 9. The deflection of the outer side of the impeller’s suction mouth was considered for the test, and the numerical oscillation results were obtained from the exact same position. Furthermore, because the coupled simulation results for the impeller deflection were obtained under a rotating coordinate frame fixed to the impeller, shown in Fig. 9 as the x and c directions, a specific formula was used to transform the calculated deflection results from the rotating frame to a stationary frame in order to compare the numerical and experimental results. The rotational position of the impeller is defined by the angle j between the positive c axis of the rotating coordinate frame and the positive y axis of the stationary coordinate frame, and j ¼ 01 when the trailing edge of the impeller is at the top position, as shown in Fig. 9. The calibration was conducted with the exact same material as that of the test impeller, and the sensor itself as well as the other components of the entire measuring chain was used for calibration. The groups of 96 J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 Fig. 9. Sensor arrangement and coordinate system. Fig. 10. Calibration curve of transient pressure sensor. calibration points and corresponding linear trend lines are shown in Fig. 11, and a nearly linear relation can be observed between the output signal of the chain and the distance of the sensor from the outer impeller suction. Within the calibration range of 0.8–2.8 mm of the chain, only the part of the range approximately at the middle position can be used for the two directions. The distance between the sensors and stationary impeller at Sx,0, Sy,0 was 1.6 mm, and the change in the distance between the sensor and impeller, DS, was 0.5 mm during the test. Therefore, regardless of whether the distance was lesser or greater than Sx,y,0, a sufficient measurement range was available. A PC-driven electronic data acquisition card with 12 simultaneous-capture analog input channels and 225 kHz per channel with a 16-bit resolution for analog acquisition was utilized. The system accuracy was 0.01%. The analog signal from the sensor was then converted to a digital signal, saved and processed by the computer. The acquisition and processing procedures were controlled by a program created in the LabVIEW graphical programming language. The program includes instructions for data acquisition, data read, array converting, filter, and display blocks, and it can directly call the bottom-level driver files of the card to finish the measurement. Signal acquisition, reading, processing, saving, and display can be performed simultaneously with the system. 2.4. Measurement procedure The oscillations of the rotor were measured under a stationary coordinate frame. Sx and Sy in Fig. 9 are the measured distance values between the sensors and the impeller when the pump was running at the test speed. The initial distance between the sensor and the outer impeller suction in the x direction, Sx,0, was considered to be a zero for the rotor oscillation when pump was not running; the same for Sy,0 in the y direction. The rotor deflection values, therefore, can be obtained in two directions using Eqs. (3) and (4). DSx ¼ Sx Sx,0 , ð3Þ DSy ¼ Sy,0 Sy : ð4Þ J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 97 Fig. 11. Calibration curves of proximity sensors. Several steps were included in the measurement procedure: the ‘‘slow rotation without water (slow-dry)’’ measurement to register the existing bearing clearances and the deviations from the roundness of the outer impeller suction diameter, the ‘‘test speed without water (dry)’’ measurement to confirm that the mechanical balance of the rotor was sufficient; and the ‘‘test speed with water (wet)’’ measurement. Details about the three steps can be found in the former work by Benra (2006). To obtain the test oscillation value induced by the hydraulic force, the possible mechanical rest unbalances and the deviations from the roundness of the outer impeller suction were subtracted from the wet measurement. The test oscillation value can be calculated by DSx,test ¼ ðSx,wet Sx,0 ÞðSx,dry Sx,0 Þ ¼ Sx,wet Sx,dry , ð5Þ DSy,test ¼ ðSy,0 Sy,wet ÞðSy,0 Sy,dry Þ ¼ Sy,dry Sy,wet : ð6Þ The initial distances of Sx,0 and Sy,0 had no exact values but a range of values due to the deviations in the roundness of the rotor; consistently they were eliminated in the above formulas with no practical implications. Note that the ranges of Sx,0 and Sy,0 were at the appropriate position in the calibration range of the system because Sx and Sy should always be in the calibration range during the test. The test oscillation values in the two directions for every monitored time point, therefore, can be calculated directly from the measured distance values when the pump was running at test speed under dry and wet conditions. Finally, the x direction and y direction signals obtained separately by two sensors in two directions compose the oscillation curves in the xy plane. The calculated deflections for the x and y coordinates can be plotted as functions of the impeller turning angle, j, for all investigated speeds of the impeller and all volume flow rates. To eliminate the uncertainty and error during the measurement and to obtain the periodic results, a phase-averaged procedure that considers the results of the deflection or transient pressure for 50 rotations was performed for every operational condition. A Butterworth low-pass filter block without a phase shift was created. The input signal was filtered in the forward direction, and then the filtered sequence was reversed and reprocessed through the filter, which doubles the filter order and ensures a zero-phase distortion. The zero-phase shift function ensures that less phase errors would be caused during the data acquisition. The cut-off frequency was set to 50 Hz for the oscillation and 500 Hz for the transient pressure measurement, and according to the blade passing frequency (max. 24 Hz), the Nyquist–Shannon sampling theorem was employed to determine the dominant physical phenomenon. 3. Results and discussions 3.1. Characteristic curve of the test pump A comparison of the head curves obtained from numerical calculations and experiments for all operational conditions of nominal speed and other rotation speeds is shown in Fig. 12. For the CFD results, the head was obtained by averaging the values during the last impeller rotation since these values fluctuate with the relative impeller positions. For the measurement procedure, signals were obtained by two static pressure sensors at the inlet and outlet, and the head value was evaluated by averaging the values from tens of impeller rotations. Therefore, signal noise and other errors were eliminated to obtain a stable delivery head value. For the nominal speed, the numerical simulation result was somewhat lower for the part-load conditions and higher for the over-load condition than the corresponding experimental results, and the best agreement appears at the point with the designed flow rate. The agreement at the over-load operating points was better than that at the part-load operating points. For n ¼1250 rpm, the predicted head result and test result exhibit good agreement when Q¼22 l/s, and the simulated head result was lower than the experimental result when Q¼11 l/s. For the flow rates of 1000 rpm and 800 rpm, the calculated results exhibit good agreement with the measured results. 98 J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 Fig. 12. Calculated and measured characteristic curves. Fig. 13. Radial hydrodynamic forces calculated by two-way coupling method for n ¼1440 rpm: (a) forces in rotating coordinates for Q¼ 42 l/s and (b) forces in stationary coordinates for all examined flow rates. 3.2. Hydrodynamic force with two-way coupling In Figs. 13 and 14, the hydrodynamic forces acting on the impeller surface for different operational conditions are analyzed considering the FSI effect, which means that both the impeller deformation and fluid redistribution influences were considered using the staggered two-way coupling calculation method. Theoretically, the force predicted by the FSI method should be more accurate and closer to reality than that obtained without considering FSI. The time-dependent hydrodynamic force is the direct cause of oscillations of the single-blade pump impeller. Both normal and tangential stresses exist on the surface of the impeller in a flow field. The tangential stress caused by friction forces of the flow field is much smaller than the normal stress caused by fluid pressure, consistently the pressure force was considered in this study for impeller oscillation and FSI calculation. In addition, only radial oscillation of the impeller shaft and hydrodynamic force in the x and y directions were investigated. For each volume element of the fluid mesh on the wetted surfaces of the pump impeller, the calculated pressure force can be decomposed into an x-component and a y-component, and the summation over all impeller surface elements in both directions yields the forces affecting the impeller rotor in each time step. Furthermore, because the impeller is fixed to the shaft, the resultant forces from all volume elements on the impeller surface can be regarded as the force acting on the axial line of the rotor shaft, and the resultant torque moment and bending moment of these forces, therefore, exist and can be transferred to the impeller structure during the two-way coupling calculation. The transient pressure force acting on the impeller can be also regarded as the sum of the forces on blade, hub, and shroud, and the results of these force components for an impeller rotation under a specific condition (n ¼1440 rpm, Q¼42 l/s) are shown in Fig. 13(a) as an example. The right-angled coordinate system rotating with the impeller defined above is used to show the hydrodynamic force distribution in the rotating frame. Some nonphysical oscillations with small amplitude of the force curves can be clearly observed for the blade and shroud force components, and therefore, the curve of the resultant force is affected by these oscillations. A comparison with the force results obtained by only CFD calculation indicates that these nonphysical oscillations were caused by the two-way coupling strategy in the CFD and CSD calculation procedure and that the staggered load transfer iteration can affect the stability of the FSI calculation system. The stability J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 99 Fig. 14. Radial hydrodynamic forces in stationary coordinates calculated by two-way coupling method for low rotation speed: (a) n¼1250 rpm and (b) n¼ 1000 rpm. of the FSI partitioned method is important for obtaining reasonable results. Compared with the deflection results of the impeller structure in the same calculation system, the hydrodynamic force results calculated by the two-way coupling method are more sensitive to the stability of the partitioned FSI calculation strategy. However, the nonphysical oscillations of the force curves are still in the acceptable range, and the force results are relatively stable. In addition, the pressure forces on the blade and shroud are in the second quadrant, and the force on the blade is clearly larger than the forces acting on the shroud. The force on the hub is located in the third quadrant and is the smallest of the three force components because, compared with the forces on the blade and shroud, the acting surface portion of the hub force, whose projection on the machine axis is a non-zero value, is the smallest. Furthermore, the c-component of the force on the hub is directed against the c-components of the blade and shroud forces. The radial forces in the absolute coordinate system for all of the examined rotation speeds and flow rates were computed, and they are function of the impeller rotation angle, j, as shown in Figs. 13(b) and 14. The arrow indicates the force at the initial impeller position ðj ¼ 01Þ, which rotates in the counterclockwise direction with an increasing flow rate for each rotation speed. The forces at the initial positions of all examined operational points are in the second quadrant, and the force vectors rotate in the clockwise direction with the impeller rotation. The nonphysical oscillations of the force curves caused by the two-way coupling calculation can also be observed in the absolute coordinate system, but the force curves are still circles or ellipses. In addition, the forces in the second and third quadrants increased significantly with the flow rate for every rotation speed. To determine the magnitude of the radial force, the forces were divided by a reference force according to Eq. (7). Fref ¼ r 2 u22 b2 r 2 : ð7Þ Therefore, dimensionless forces were obtained, and the reference force had distinct values for different rotation speeds. In addition, different maximal flow rates were obtained when the valve was wide open for three rotation speeds (42 l/s for 1440 rpm, 35 l/s for 1250 rpm, and 27 l/s for 1000 rpm), but the shapes of the dimensionless force curves for the maximal flow rates of different rotation speeds are nearly the same. Moreover, even when the nonphysical oscillations from twoway coupling are not considered, the amplitudes have a few differences as well. Furthermore, for different rotation speeds, a Q of 22 l/s represents a different percentage of the maximal flow rate—52.4% for 1440 rpm, 62.9% for 1250 rpm, and 81.5% for 1000 rpm—and the dimensionless force values on the negative axis in the x direction are approximately 0.7, 0.8, and 1, respectively. The absolute values increase by increasing the examined flow rate percentage of the maximal flow rates, which is done by increasing the rotation speed. The same trend was found for Q¼11 l/s. Therefore, the percentage of maximal flow rate that is represented by a certain flow rate at different rotation speeds is important for evaluating the dimensionless force at these different rotation speeds. To identify the importance of the vibration, comparisons of the hydrodynamic force results calculated with and without the FSI effect for a nominal rotation speed are shown in Figs. 15–17. The force amplitude calculated without FSI is smaller, but a small difference can be observed for the examined conditions. A more obvious difference appears for the condition in which Q¼42 l/s because the vibration displacement is relatively large. The phase difference can be observed for Q¼42 l/s and Q¼ 11 l/s. Compared with the FSI results, a phase lead is obtained for results without FSI under an over-load condition, and a phase lag is observed for the condition in which Q¼11 l/s. In addition, because the force results without FSI are not affected by the deformation of the impeller structure, smoother hydrodynamic force curves are obtained. 100 J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 Fig. 15. Comparison between hydrodynamic forces with and without FSI for Q¼ 42 l/s, n ¼1440 rpm. Fig. 16. Comparison between hydrodynamic forces with and without FSI for Q¼ 22 l/s, n ¼1440 rpm. Fig. 17. Comparison between hydrodynamic forces with and without FSI for Q¼ 11 l/s, n ¼1440 rpm. 3.3. Rotor oscillations due to hydrodynamic forces The comparisons of numerical deflection results considering strong FSI and the measurement results for flow rates with a wide open valve and a Q of 22 l/s for different rotation speeds are shown in Figs. 18–20. Both the 2D rotor oscillation orbit curves in the xy plane and the 1D component curves of the impeller deflection orbit in two directions are considered. Not only the amplitude of oscillation but also the phase of the oscillation curve was investigated. All the calculated and measured curves are presented versus the impeller rotation angle j for one impeller rotation, and j is also a function of time. For the nominal rotation speed of 1440 rpm, the calculated and measured results are compared under the over-load condition where Q¼42 l/s and the part-load conditions where Q¼22 l/s. For the over-load condition, shown in Fig. 18(a) and (b), the measured and calculated deflection curves coincide well for the entire fourth quadrant and half of the range of the first and third quadrants. The FSI calculated amplitude was smaller in the rest of the range, not only for this condition but also for other analyzed conditions. In addition, the arrows show the deflection vectors at an impeller rotational J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 101 Fig. 18. Comparisons of FSI calculated and rotor deflections measured for n¼ 1440 rpm: (a) orbit curves for Q¼ 42 l/s, (b) x, y component curves for Q¼ 42 l/s, (c) orbit curves for Q ¼22 l/s, and (d) x, y component curves for Q ¼22 l/s. Fig. 19. Comparisons of FSI calculated and rotor deflections measured for n¼ 1250 rpm: (a) orbit curves for Q¼ 35 l/s, (b) x, y component curves for Q¼ 35 l/s, (c) orbit curves for Q ¼22 l/s, and (d) x, y component curves for Q ¼22 l/s. 102 J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 Fig. 20. Comparisons of FSI calculated and rotor deflections measured for n ¼1000 rpm: (a) orbit curves for Q¼ 27 l/s, (b) x, y component curves for Q ¼27 l/s, (c) orbit curves for Q¼22 l/s, and (d) x, y component curves for Q¼ 22 l/s. position, j of 01 for both numerical and measured results, and the initial deflection positions are in the second quadrant for this and all other operational conditions in this study. Meanwhile, the phases of the two initial positions exhibit a rather good agreement, indicating that the two-way coupling calculation obtained a good phase prediction of the impeller oscillation curve for the over-load condition. From the results of the x-component and y-component of the deflection orbit, an obvious difference between the numerical and measured results can be observed in a specific j range. The FSI results were overestimated for the positive x direction in the range of approximately 0–1601 and were underestimated for the forward y direction in the range of 0–1801. A similar trend can be observed for all the concerned off-design conditions, except that the specific j range is slightly different. Furthermore, for the y-component curve, the numerical results exhibit a rather good agreement with the experimental results for the second half of the impeller rotation, and the peak-to-peak value between the x-component and y-component curves is nearly 100–1201 for all conditions. For the part-load condition where Q¼22 l/s, shown in Fig. 18(c) and (d), an acceptable agreement between the numerical and experimental results can be observed in a quadrant range similar to that for the over-load condition. The initial deflection vector of the numerical result has a small phase lag compared to the measured result. For the x-component result, an obvious difference between the numerical and experimental results can be observed in the j range of approximately 300–3601. Compared with the 1440-rpm rotation speed, the 1250-rpm and 1000-rpm rotation speeds exhibit similar relation between the numerical and experimental rotor deflection orbit curves for the corresponding flow rate, shown in Figs. 19 and 20. Thus, for the maximal flow rate condition of each rotation speed, Q¼35 l/s for 1250 rpm or Q¼27 l/s for 1000 rpm, the calculated and measured deflection orbit curves have a similar relation as that in the case of 42 l/s for 1440 rpm. The results for Q¼22 l/s at 1250 rpm and 1000 rpm are nearly the same as those for Q of 22 l/s at 1440 rpm. Overall, the only obvious difference between the results for the low rotation speeds compared and those for the nominal speed is in terms of the amplitude of deflection for the specified flow rate, and the amplitude decreases with the rotation speed. In general, the behavior of the FSI two-way coupling results is confirmed by the experiment, but the measurements show some deflections for about half of an impeller rotation. On an analysis of the angles of the obtained x and y components, obvious deviations are observed in the first half of the impeller rotation, a j range from approximately 01 to 1801, for both the x and y directions under all the operational conditions and rotation speeds considered. Furthermore, to determine the cause of the deviation between the FSI simulation and experimental deflections, the transient pressures for several operational conditions at the casing obtained by both the CFD and experimental methods are qualitatively analyzed, and a few examples are shown in Fig. 21. The main difference between the CFD and measured transient pressures can be clearly observed in the j range of 01 to approximately 1801, which is approximately the same range in J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 103 Fig. 21. Comparisons of CFD calculated and transient pressure measured for (a) Q ¼42 l/s, n ¼1440 rpm, (b) Q¼ 22 l/s, n¼ 1250 rpm, (c) Q¼27 l/s, n¼ 1000 rpm, and (d) Q¼ 22 l/s, n¼ 1000 rpm. which a difference exists between the FSI and measurement results for rotor deflection. This indicates that in this j range, the CFD results cannot capture all of the features of the transient pressure in the pump casing, resulting in an obvious difference between the calculated and measured deflection orbit. The accuracy limitation of the CFD simulation conducted in this study for dealing with an asymmetrical fluid field, as in the case of a single-blade pump impeller, is the main reason for this difference. Because the transient pressure forces transferred to the pump structure are the most important values for the FSI calculation, the CFD calculation is an essential part of the FSI solution system. Therefore, the difference between the CFD and experimental transient pressures has an important effect on the rotor deflection prediction. Future research should focus on determining the reason for the CFD prediction deviation. Although several reasons for the deviation of numerical and experimental results have been discussed in former work, other possible causes should be considered and further investigated: (1) The dynamic characteristic (deformation) of the bearings during the pump operation was not considered. The bearing was assumed as a rigid body, and a fixed support was used during the FSI investigation. (2) The complex flow behavior in the seal during the pump operation was not considered. Furthermore, the flow in the seal was different for the ‘‘wet’’ and ‘‘dry’’ operations while obtaining the hydrodynamic rotor deflection. (3) The vibration behaviors of the casing were different for the ‘‘wet’’ and ‘‘dry’’ conditions, which would affect the position of the proximity sensor relative to the rotor because the sensors were mounted on the casing. (4) The alignment error of the shaft coupling was not considered during the FSI calculation. (5) The convergence criterion used was 0.01 for the load transfer process of the FSI partitioned calculation, which is the default value. A stricter criterion may be used to obtain more precise FSI results. 4. Conclusions This step advances the methods used in previous studies. In the present study, the periodically unsteady flow-induced impeller oscillations for a single-blade pump were investigated both numerically by strong two-way coupling FSI simulations and experimentally by non-contact deflection measurements under off-design conditions. Comparisons between the FSI simulation and deflection measurement results, considering both phase and amplitude, were conducted for different flow rates and rotation speeds. The initial deflection positions were in the second quadrant, and the phases of the two initial positions exhibited a rather good agreement for the maximal flow rate of each rotation speed, while a small phase lag was observed for the other examined conditions. The amplitude results were confirmed by the experiment, but the measurements exhibited some deflections for approximately half of an impeller rotation. In the x and y component angle results, obvious deviations were observed in the first half of the impeller rotation, that is, the j range from approximately 01 to 1801 for both the x and y directions, and an obvious difference was observed in the j range of approximately 300–3601 for the x direction only under the conditions in which Q¼22 l/s. The peak-to-peak phase value between the x-component and y-component curves was nearly 100–1201 for all the conditions. Furthermore, the amplitude of the deflection orbit curves for the specified flow rate (22 l/s, for instance) decreased with the rotation 104 J. Pei et al. / Journal of Fluids and Structures 35 (2012) 89–104 speed. To determine the cause of the deviation, the transient pressures at the casing obtained by both the CFD and experimental methods were qualitatively analyzed. The difference between the CFD and experimental transient pressures had an important effect on the rotor deflection prediction in the j range of 01 to approximately 1801. Hydrodynamic forces were analyzed considering the FSI effect. The pressure forces on the blade and shroud were in the second quadrant, and the force on the hub was located in the third quadrant in the rotating coordinate frame. For dimensionless forces in an absolute coordinate frame, the shapes and amplitudes of the dimensionless force curves for maximal flow rates of different rotation speeds were nearly the same, and the amplitude increased by increasing the examined flow rate percentage of the maximal flow rates, which was done by increasing the rotation speed. Finally, additional possible causes of the deviation between the FSI calculation and experimental rotor oscillation results were discussed. A more accurate CFD calculation for the fluid field in the single-blade pump should be conducted in future research as the basis for more accurate FSI simulations. Acknowledgements This research is supported by the National Outstanding Young Scientists Foundation of China (Grant no. 50825902), Jiangsu Provincial Project for Innovative Postgraduates of China (Grant no. CX10B_262Z), and Natural Science Foundation of Jiangsu Province (Grant no. BK2010347). 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