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D. Kumar, K. Murugesan, H. Thomas (2008)
Numerical Simulation of Double Diffusive Mixed Convection in a Lid-Driven Square Cavity Using Velocity-Vorticity FormulationNumerical Heat Transfer, Part A: Applications, 54
Nagendra Dittakavi, Aditya Chunekar, S. Frankel (2010)
Large Eddy Simulation of Turbulent-Cavitation Interactions in a Venturi NozzleJournal of Fluids Engineering-transactions of The Asme, 132
Deyou Li, Hongjie Wang, Yonglin Qin, Han Lei, Xianzhu Wei, D. Qin (2017)
Entropy production analysis of hysteresis characteristic of a pump-turbine modelEnergy Conversion and Management, 149
Yuning Zhang, Kai-hua Liu, H. Xian, Xiaoze Du (2018)
A review of methods for vortex identification in hydroturbinesRenewable & Sustainable Energy Reviews, 81
LU Fu-An, Xuejun Wang, Qi Da-tong, Cai Jian-cheng (2010)
Study of the Tonal Casing Noise of a Centrifugal Fan at the Blade Passing FrequencyJournal of Low Frequency Noise, Vibration and Active Control, 30
Yuning Zhang, Ting Chen, Jinwei Li, Ji-xing Yu (2017)
Experimental Study of Load Variations on Pressure Fluctuations in a Prototype Reversible Pump Turbine in Generating ModeJournal of Fluids Engineering-transactions of The Asme, 139
Ning Zhang, Minguan Yang, B. Gao, Z. Li, Dan Ni (2016)
Investigation of Rotor-Stator Interaction and Flow Unsteadiness in a Low Specific Speed Centrifugal PumpStrojniski Vestnik-journal of Mechanical Engineering, 62
Q. Le, J. Franc, J. Michel (1993)
Partial Cavities: Global Behavior and Mean Pressure DistributionJournal of Fluids Engineering-transactions of The Asme, 115
(2017)
Numerical investigation on the unsteady characteristics of reactor coolant pumps with nonuniform inflow
Q. Liu, K. Yang, D. Li, R. Gong (2013)
Research of fluid-induced pressure fluctuation due to impeller-volute interaction in a centrifugal pumpIOP Conference Series: Materials Science and Engineering, 52
R. Barrio, E. Blanco, J. Parrondo, J. González, Joaquín Fernandez (2008)
The Effect of Impeller Cutback on the Fluid-Dynamic Pulsations and Load at the Blade-Passing Frequency in a Centrifugal PumpJournal of Fluids Engineering-transactions of The Asme, 130
Guanghao Chen, Guo-yu Wang, Changli Hu, B. Huang, Yuan Gao, Mindi Zhang (2015)
Combined experimental and computational investigation of cavitation evolution and excited pressure fluctuation in a convergent–divergent channelInternational Journal of Multiphase Flow, 72
Toshiya Kimura, Y. Yoshida, Tomoyuki Hashimoto, Mitsuru Shimagaki (2008)
Numerical Simulation for Vortex Structure in a Turbopump Inducer: Close Relationship With Appearance of Cavitation InstabilitiesJournal of Fluids Engineering-transactions of The Asme, 130
J. Pei, Wenjie Wang, S. Yuan (2014)
Statistical analysis of pressure fluctuations during unsteady flow for low-specific-speed centrifugal pumpsJournal of Central South University, 21
X. Gu, Jianfei Song, Yaodong Wei (2016)
Experimental study of pressure fluctuation in a gas-solid cyclone separatorPowder Technology, 299
R. Arndt (2002)
CAVITATION IN VORTICAL FLOWSAnnual Review of Fluid Mechanics, 34
X. Su, Simin Huang, Xuejiao Zhang, Sun-sheng Yang (2016)
Numerical research on unsteady flow rate characteristics of pump as turbineRenewable Energy, 94
B. Gao, Ning Zhang, Z. Li, Dan Ni, Minguan Yang (2016)
Influence of the Blade Trailing Edge Profile on the Performance and Unsteady Pressure Pulsations in a Low Specific Speed Centrifugal PumpJournal of Fluids Engineering-transactions of The Asme, 138
(2013)
Measurements of internal flow in centrifugal pump at part-load conditions based on LDV
L. Yun, Wang Dezhong, Yin Junlian, Hu Yaoyu, Ran Hongjuan (2017)
Numerical investigation on the unsteady characteristics of reactor coolant pumps with non-uniform inflowNuclear Engineering and Design, 320
Jiezhi Wu, Hui-yang Ma, M. Zhou (2006)
Vorticity and Vortex Dynamics
Cuizhi Gao, Guogang Sun, R. Dong, S. Fu (2010)
Characterizing the dynamic property of the vortex tail in a gas cyclone by wall pressure measurementsFuel Processing Technology, 91
Cai Jian-cheng, Qi Da-tong, LU Fu-An, X. Wen (2010)
Study of the Tonal Casing Noise of a Centrifugal Fan at the Blade Passing Frequency. Part I. AeroacousticsJournal of Low Frequency Noise, Vibration and Active Control, 29
裴吉, 王文杰, 袁寿其 (2014)
Statistical analysis of pressure fluctuations during unsteady flow for low-specific-speed centrifugal pumps, 21
Xianwu Luo, A. Yu, B. Ji, Yulin Wu, Y. Tsujimoto (2017)
Unsteady vortical flow simulation in a Francis turbine with special emphasis on vortex rope behavior and pressure fluctuation alleviationProceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 231
J. Pei, S. Yuan, Xiaojun Li, Jianping Yuan (2014)
Numerical prediction of 3-D periodic flow unsteadiness in a centrifugal pump under part-load conditionJournal of Hydrodynamics, 26
J. Pei, H. Dohmen, S. Yuan, F. Benra (2012)
Investigation of unsteady flow-induced impeller oscillations of a single-blade pump under off-design conditionsJournal of Fluids and Structures, 35
S. Salehi, M. Raisee, Michel Cervantes, A. Nourbakhsh (2018)
On the flow field and performance of a centrifugal pump under operational and geometrical uncertaintiesApplied Mathematical Modelling
R. Huang, B. Ji, Xianwu Luo, Z. Zhai, Jia-jian Zhou (2015)
Numerical investigation of cavitation-vortex interaction in a mixed-flow waterjet pumpJournal of Mechanical Science and Technology, 29
Hucan Hou, Yongxue Zhang, Zhenlin Li, Y. Zhang (2017)
A CFD study of IGV vane number on hydraulic characteristics and pressure pulsation of an is centrifugal pumpJournal of Vibroengineering, 19
B. Ji, Xianwu Luo, R. Arndt, Yulin Wu (2014)
Numerical simulation of three dimensional cavitation shedding dynamics with special emphasis on cavitation–vortex interactionOcean Engineering, 87
Xinglin Yang, Chenhui Wu, Huabing Wen, Linglong Zhang (2018)
Numerical simulation and experimental research on the aerodynamic performance of large marine axial flow fan with a perforated bladeJournal of Low Frequency Noise, Vibration and Active Control, 37
Xiaoqi Jia, B. Cui, Zuchao Zhu, Yu‐Liang Zhang (2017)
Experimental Investigation of Pressure Fluctuations on Inner Wall of a Centrifugal PumpInternational Journal of Turbo & Jet-Engines, 36
E. Foeth (2008)
The structure of three-dimensional sheet cavitation
B. Ji, Xianwu Luo, Yulin Wu, K. Miyagawa (2014)
Numerical investigation of three-dimensional cavitation evolution and excited pressure fluctuations around a twisted hydrofoilJournal of Mechanical Science and Technology, 28
B. Zhu, Hongxun Chen (2012)
Cavitating Suppression of Low Specific Speed Centrifugal Pump with Gap Drainage BladesJournal of Hydrodynamics, 24
T. Lei, Z. Shan, C. Liang, Wang Chuan, W. Bin (2014)
Numerical simulation of unsteady cavitation flow in a centrifugal pump at off-design conditionsProceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 228
Hanshin Seol (2013)
Time domain method for the prediction of pressure fluctuation induced by propeller sheet cavitation: Numerical simulations and experimental validationOcean Engineering, 72
R. Barrio, Joaquín Fernandez, E. Blanco, J. Parrondo (2011)
Estimation of radial load in centrifugal pumps using computational fluid dynamicsEuropean Journal of Mechanics B-fluids, 30
Jingmei Qiu, Chen-Jun Yang, Xiaoqian Dong, Zongbao Wang, Wei Li, F. Noblesse (2018)
Numerical Simulation and Uncertainty Analysis of an Axial-Flow Waterjet PumpJournal of Marine Science and Engineering
G. Lu, Z. Zuo, Yuekun Sun, De-min Liu, Y. Tsujimoto, Shuhong Liu (2017)
Experimental evidence of cavitation influences on the positive slope on the pump performance curve of a low specific speed model pump-turbineRenewable Energy, 113
E. Branlard (2017)
Wind Turbine Aerodynamics and Vorticity-Based Methods: Fundamentals and Recent Applications
Chuan Wang, W. Shi, Xikun Wang, Xiaoping Jiang, Yang Yang, Wei Li, Lingjiu Zhou (2017)
Optimal design of multistage centrifugal pump based on the combined energy loss model and computational fluid dynamicsApplied Energy, 187
Anup Kc, Young-Ho Lee, B. Thapa (2016)
CFD study on prediction of vortex shedding in draft tube of Francis turbine and vortex control techniquesRenewable Energy, 86
Yehui Zhang, Yongxue Zhang, Jinya Zhang, Hucan Hou (2014)
Study on Pressure Pulsation in the Volute of a Centrifugal Pump by Large Eddy Simulation
B. Huang, Yu Zhao, Guo-yu Wang (2014)
Large Eddy Simulation of turbulent vortex-cavitation interactions in transient sheet/cloud cavitating flowsComputers & Fluids, 92
To provide a comprehensive understanding of the pressure fluctuation–vortex interaction in non-cavitation and cavita- tion flow, in this article, the unsteady flow in an ultra-low specific-speed centrifugal pump was investigated by numerical simulation. The uncertainty of the numerical framework with three sets of successively refined mesh was verified and validated by a level of 1% of the experimental results. Then, the unsteady results indicate that the features of the internal flow and the pressure fluctuation were accurately captured in accordance with the closed-loop experimental results. The detailed pressure fluctuation at 16 monitoring points and the monitoring of the vorticity suggest that some inconsistent transient phenomena in frequency spectrums show strong correlation with the evolution of vortex, such as abnormal increasing amplitudes at the monitoring points near to the leading edge on the suction surface and the trailing edge on the pressure surface in the case of lower pressurization capacity of impeller after cavitation. Further analysis applies the relative vortex transport equation to intuitionally illustrate the pressure fluctuation–vortex inter- action by the contribution of baroclinic torque, viscous diffusion and vortex convection terms. It reveals that the effect of viscous diffusion is weak when the Reynolds number is much greater than 1. Pressure fluctuation amplitude enlarges on the suction side of blade near to the leading edge due to the baroclinic torque in cavitation regions, whereas the abnormal increase of pressure fluctuation after cavitation on the pressure surface of blade approaching the trailing edge results from the vortex convection during vortices moving downstream with the decrease of available net positive suction head at the same instance. Keywords Unsteady cavitation flow, an ultra-low specific-speed centrifugal pump, pressure fluctuation, the vortex transport equa- tion, numerical simulation Introduction The specific speed, n , of an ultra-low specific-speed centrifugal pump is less than 30, and thus impeller passages are typically designed to be narrow and long to satisfy low flow rate and high head requirements. However, there 1 2 are certain operating problems, namely, low-capacity performance, hump under partial flow rate, overloaded 3 4 likelihood and cavitation, which are difficult to deal with. Moreover, there is typically a large adverse pressure gradient accompanied by separation vortices in impeller passages. College of Mechanical and Transportation Engineering, China University of Petroleum, Beijing, China Beijing Key Laboratory of Process Fluid Filtration and Separation, China University of Petroleum, Beijing, China Sinopec Engineering Incorporation Ltd., Beijing, China Beijing Aerospace Petrochemical Technology & Equipment Engineering Corporation Ltd., Beijing, China Corresponding author: Yongxue Zhang, 18 Fuxue Road, Changping, Beijing, China. Email: zhyx@cup.edu.cn Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www. creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 528 Journal of Low Frequency Noise, Vibration and Active Control 38(2) 5,6 The monitoring of complex inner flow phenomena in the hydraulic machinery has been a hot topic. The detection and diagnosis of the inception and the severity of cavitation provide means for preventing accidents. When severe cavitation occurs, it is usually accompanied by significant noise. However, sometimes no audible noise will be heard at the cavitation inception. In order to clarify the relationship between cavitation evolution 7 8 and pressure fluctuation, Seol and Chen et al. launched the experimental and computational work on a propeller and in a convergent–divergent channel. The results showed that the evolution of cavitation can be predicted by the variation of pressure fluctuation. Ji et al. also tried to confirm the mechanisms between the two mentioned above. Through analyzing the synchronous variation between cavity volume and pressure fluctuation in cavitation zone, they came to the conclusion that the acceleration of cavity volume was the dominant source of the excited pressure, which in turn explained that pressure fluctuation could be employed to describe the development of cavitation. Jia et al. carried out dynamic pressure tests in a centrifugal pump to investigate the pressure distri- bution on volute and front shroud. The results showed that pressure fluctuation was sensitive to the flow rate, and therefore they presented the asymmetry in the pressure distribution as a function of the flow rate afterwards. Liu et al. studied the characteristics of pressure fluctuation at the impeller outlet. The results presented that the main component of the impeller–volute interaction was the blade passing frequency, and the harmonic interference was due to the unsteady flow. Tan et al. investigated the unsteady cavitation flow in a centrifugal pump under off- design conditions. The research indicated that cavitation has more influence on pressure fluctuation in the impeller under partial discharge than large discharge owing to the more intense disturbances caused by cavitation shedding and explosion under partial discharge. Zhang et al. investigated the frequency domain characteristics of pressure fluctuation with a wide range of the low-load conditions in a prototype reversible pump turbine. The results revealed that the propagation of pressure fluctuation at vaneless space led to a significant amplitude to the upstream and a smaller one to the downstream areas, which reflected the influence mechanism of rotor–stator interactions. The research of Zhang et al. in our research group took the form of large eddy simulation to calculate the pressure pulsation in the volute of a low specific-speed centrifugal pump. It helped to understand the variation of pressure pulsation with different flow rates, radial distance, circumferential angles and axial distance. In light of the effect of the inlet guide vane number on hydraulic stability, Hou et al. selected the inlet guide vane with six vanes for engineering application after analyzing pressure fluctuation amplitude and power spec- trum density. In previous studies, many scholars have found that the evolution of cavitation had a close relationship with the change of the vortex field. Arndt pointed out that vortex cavitation had a dominant effect on the inception process in a broad range of turbulent flow, and vortex instability was associated with sheet/cloud cavitation. Le et al. researched the characteristics of flow under partial cavitation. The results indicated that the instability of cavitation had an intimate contact with the thickness of the cavity and the reentrant jet, which further led to vortices production. Kimura and Yoshida conducted an unsteady simulation in a turbopump inducer under non-cavitation conditions to study the vortex structure with different flow rates. They found that the change regulation of the vortex structure was analogous to that of rotating cavitation, which strongly demonstrated that the vortex was responsible for the occurrence of rotating cavitation. Using different twisted hydrofoils, Ji et al. and Huang et al. investigated the cavitation–vortex interaction. They found that cavitation could strengthen vortex production and increase the boundary layer thickness with local separation and flow unsteadiness. Furthermore, they revealed the influence of cavitation on the vorticity field by employing the vortex transport equation. Subsequently, Huang et al. studied the structure of cavitation flow in a mixed-flow waterjet pump. Considering the rotation effect, they used the relative transport equation to improve the understanding of mechanisms of the cavitation–vortex interaction, and then he drew conclusions that vortex dilation and stretching terms were the main contributors as cavitation occurred and diffused. Although great progress has been made on the instability flow in fluid machinery by numerical and experi- mental methods, previous researches often focused on the analysis of pressure fluctuation or the investigation of internal flow field. There is less attention that was paid to the interaction between the acoustic signal (pressure fluctuation) and the vortex structure (the internal flow field). Indeed, some special pressure pulsation phenomena are closely related to the evolution of the vorticity field, so the pressure fluctuation–vortex interaction is a worthwhile topic. Anup et al. utilized a numerical analysis to investigate the pressure oscillation owing to the 23 24 vortex shedding under partial discharge in a Francis turbine. Gao et al. and Gu et al. carried out experiments of pressure fluctuation measurements to get further insight into dynamic properties of the vortex in different cyclone separators. Luo et al. studied the variation of pressure fluctuation caused by vortex rope under different gas injection, and the air occupation at the draft tube center for the stable operation was proposed. However, certain unusual characteristics of pressure fluctuation induced by vortices are still to be solved urgently in the Wang et al. 529 ultra-low specific-speed centrifugal pump with a complex vorticity field. Gao et al. demonstrated the effect of different blade trailing edge profiles on the performance and unsteady pressure fluctuation in a low specific-speed centrifugal pump. Through the vorticity distributions with different blade trailing edge profiles, they concluded that the vortex shedding intensity from the blade trailing edge was the dominated factor for the change of the performance and pressure fluctuation amplitude. Unfortunately, the effect of the vortex evolution over time on the pressure fluctuation could not be captured owing to the mesh resolution at the blade trailing edge region. However, the vortex transport equation is a powerful and available mean to quantitatively analyze characteristics and rules of the vortex distribution in the viscous flow. Branlard systematically derived the vortex transport equation, and then he applied the vortex-based model to the wind turbine aerodynamics. Thereafter, the vortex 18–20,28,29 transport equation has been widely applied for better understanding the variation of the vortex field. As a result, the vortex transport equation can be used to combine pressure fluctuation with vortex dynamics, which provides a better insight of the excitation effect of cavitation on pressure fluctuation. In this paper, a three dimensional (3D) unsteady simulation for non-cavitation and cavitation flow in the whole flow passage of an ultra-low specific-speed centrifugal pump was implemented, and the pressure fluctuation was analyzed via fast Fourier transform (FFT). Further analysis put emphasis on some special phenomena in the evolution of pressure fluctuation, such as the abnormal increase of pressure fluctuation amplitudes at certain monitoring points after cavitation, by using the derived relative vortex transport equation to clearly explain the mechanism of the pressure fluctuation–vortex interaction in an ultra-low specific-speed centrifugal pump for the first time. Experimental tests and numerical method Experimental tests Hydraulic tests were carried out to validate the accuracy of the simulation. As shown in Figure 1, an electric motor was employed to drive the centrifugal pump system with a constant rotating speed. The water supplied from a tank was returned to the same tank, namely, a closed loop. A water ring vacuum pump was installed under the tank to adjust the inlet pressure, and an electromagnetic flowmeter and two differential pressure transmitters were aiming at recording the flow and pressure conditions. The data collector was responsible for acquiring the acoustic signals of pressure fluctuation with 180 sample points per revolution and the dynamic data process program performed the FFT to obtain the information of the frequency domain. Main parameters of measuring instruments used in the experiment are listed in Table 1. Two major experiments, including cavitation perfor- mance experiments and pressure fluctuation experiments, have been done on this system. Figure 1. Schematic diagram of the hydraulic test set-up. 530 Journal of Low Frequency Noise, Vibration and Active Control 38(2) Table 1. Main parameters of the measuring instruments. Category Model Range Accuracy (%) Pressure transducer 8530B 1378.95 kPa 0.21 Torque meter CYT-302 100 Nm 0.10 Electromagnetic flowmeter EMF8301 15 m /h 0.50 Differential pressure transmitter GLP3351 0.1 to 0.2 MPa (inlet) 0.20 0 to 1.2 MPa (outlet) Figure 2. The structure of the pump. Table 2. Main geometrical parameters of the pump. Parameters Values Inlet diameter (mm) 50 Blade inlet angle ()20 Outlet diameter (mm) 32 Blade outlet width (mm) 5 Blade outlet angle ()38 Blade wrap angle ( ) 165 Physical model The ultra-low specific-speed centrifugal pump applied in the research is shown in Figure 2. It consists of a suction, a four-blade impeller, a volute and a draft tube. Considering inlet effect, the suction length is extended to five times as long as the diameter of the impeller eye, whereas the draft tube with a length five times longer than the 0:5 2pn outlet diameter of the volute is installed to prevent backflow. The specific speed, n ¼ 3:22 , is 23.3, where s 0:75 ðÞ gH Q ¼ 12.5 m /h is the design discharge and n ¼ 2900 r/min is the rotational speed. The main geometric sizes are listed in Table 2. Wang et al. 531 Figure 3. Mesh profile of the centrifugal pump. Mesh generation The full passage simulation from the inlet extension to the outlet extension was carried out with structured grids except the volute tongue (as shown in Figure 3). Considering the complex shape of cut-water, multi-block hybrid grid and refined work around the cut-water were utilized in the volute. Consequently, there are three pairs of interfaces between the cut-water and other parts of the volute. The volume average values of y were 51 and 68 for the above two regions, which both met the requirements of the simulation in the near-wall regions. Numerical model and parameter setting After cavitation, the vapor/liquid two-phase flow is supposed to be homogeneous, that is, two phases along the interface share the same velocity and pressure. The governing equations based on Reynolds-averaged Navier– Stokes method consist of mass continuity (equation (1)) and momentum (equation (2)) equations. Simulations were carried out by the commercial software ANSYS-CFX. Prior to performing a case, the re-normalization group (RNG) k–e turbulence model with scalable wall functions was applied (see Hou et al. for details). For obtaining the cavitation performance, the Zwart-Gerber-Belamri model (see Ji et al. ) was chosen. Based on the assumption of homogeneous mixture model, vapor area is relatively stable and shares the same velocity with the liquid. As the common velocity in the centrifugal pump is much lower than the local sound velocity (Mach number is less than 0.3, local sound velocity of vapor at 298 K is about 426.52 m/s), it can still be treated as an incompressible fluid @q @ þ q u ¼ 0 (1) ðÞ i @t @x @ q u @ @p @ @u @u 2 @u ðÞ i m i j k þ q u u ¼ þ l þ l þ d (2) ðÞ ðÞ m i j m t ij @t @x @x @x @x @x 3 @x i i j j i k Here, p is the pressure, u represents the velocity, l stands for the turbulent viscosity and d is the Kronecker delta t ij function. Besides, the mixture density, q , and viscosity, l , are defined as a function of vapor volume fraction m m q ¼ q a þ qðÞ 1 a (3) m v v l v l ¼ l a þ lðÞ 1 a (4) v v m v l where a is the volume fraction, the subscripts l and v indicate the liquid and vapor phases, respectively. 532 Journal of Low Frequency Noise, Vibration and Active Control 38(2) Figure 4. Positions of monitoring points in the pump. The inlet static pressure and outlet mass flow conditions were set at first, whereas all physical walls were considered as no-slip boundary conditions. The flow in impeller domain was simulated in a rotational frame, whereas the other domains were based on a stationary frame. The rotational and stationary domains were connected by the general grid interface where the grid on either side of the two connected surfaces permits 31–33 non-matching. Then, the interfaces between the rotating and stationary components were set as frozen 34,35 rotor for the steady calculation, yet transient rotor stator was set for the unsteady calculation. Finally, the calculation was regarded as convergence when the root mean square residual was below 1.0 10 . The results of the steady calculation were taken as the initial flow field in the transient flow calculation. As introduced in 2p 1 ANSYS CFX tutorials, the time-step size in the rotating machinery is calculated by , where Z ¼ 4 is the Z x number of blades, x ¼ 2pn/60 ¼ 303.69 s is the angular velocity, and thus the time-step size can be set as 5.17 10 s. However, some researches recommend that the chosen time step corresponds to the changed 36–39 angle of 3 of impeller rotation. For calculating the pressure fluctuation more accurately, the time that blades turn every 2 , which is 1.149 10 s (equivalently T/180), was chosen as the time step, where T represents the rotation period of the pump. And each time step was calculated by 10 iterations. In order to gain results close to the reality, total computing time is eight times of a period. All results and analysis of data were derived from the last six periods to guarantee the accuracy. As shown in Figure 4, 16 monitoring points were set at the middle, the pressure and the suction sides of the blade and the outlet of the volute to record pressure fluctuation in the pump. Mesh sensitivity analysis: Verification and Validation Three sets of successively refined meshes has been generated with a uniform grid refinement ratio, r =m / G iþ1 pffiffiffi m = 2 (i ¼ 0, 1, 2), where m denotes the mesh elements, and the subscript represent different mesh items. Simulation results (see Table 3) indicate that there are subtle differences when elements extend 1.50 million. For the reliability of the simulations, the uncertainty analysis on the head and efficiency were evaluated at the design discharge according to ITTC 7.5-03-01-01 recommendations. The numerical uncertainty is expressed as 2 2 2 2 2 U ¼ U þ U þ U þ U (5) SN I M T P Here, the subscripts I, M, T and P represent the uncertainties resulted from iteration, mesh density, time-step size and other parameters, respectively. In this work, the chosen time-step size was sufficiently small and the Wang et al. 533 Table 3. Head H and efficiency g versus mesh elements. Item Element Exp. Scheme 1 Scheme 2 Scheme 3 Whole passage – 1,079,609 1,521,785 2,188,330 H/H 0.995 0.988 1 1.002 g/g 1.010 0.978 1 1.003 Note: H ¼81.15 m and g ¼49.81% are the pump head and hydraulic efficiency calculated by using Scheme 2, respectively. 2 2 Figure 5. The oscillatory convergence of the head of Scheme 2 in the last period. The abscissa h indicates the angular position of a blade, and the ordinate H expresses the instantaneous head. Table 4. Iteration uncertainties of the head and efficiency in the simulation. Iteration uncertainty Scheme 1 Scheme 2 Scheme 3 Head (%D) 0.0345 0.0302 0.0289 Efficiency (%D) 0.0362 0.0321 0.0314 Note: As mentioned in Qiu et al., the verification of the mesh-density uncertainty is listed in Table 5. uncertainty of the time-step size was neglected, while the uncertainty due to other parameters U was not included and was also neglected. Thus 2 2 2 U ¼ U þ U (6) SN I M Take the oscillatory convergence of the head of Scheme 2 as an example, U is defined as U ¼ ðÞ S S (7) I U L where S and S denote the maximum and minimum values of results in the last period. Figure 5 illustrates the U L oscillatory convergence of the head of Scheme 2 in the last period. Therefore, U ¼ 0.0302%D, where D is the experimental data. Similarly, the uncertainties of the head and efficiency due to iteration are listed in Table 4. From Tables 4 and 5, the iteration uncertainties are negligibly smaller than mesh-density uncertainties, and hence U U . SN M 534 Journal of Low Frequency Noise, Vibration and Active Control 38(2) Table 5. Mesh-density uncertainties of the head and efficiency in the simulation. pffiffiffi pk lnðÞ e =e e MC FM e FM FM pffiffi Item R ¼ p ¼ d ¼ pffiffi C ¼ 2 1 U (%D) M k M RE k eMC ln 2 2 1 H 0.165 5.200 0.0316 5.063 0.357 g 0.136 5.749 0.0237 6.333 0.548 Table 6. The comparison error E and validation uncertainty U . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Comparison error E ¼ D–S Validation uncertainty U ¼ U þ U SN D Scheme Head (%D) Efficiency (%D) Head (%D) Efficiency (%D) 3 0.706 0.696 0.885 0.978 The comparison error E was evaluated by the experimental results D, and the simulated results were based on Scheme 3. The experimental uncertainty U is assumed to be 0.81%D by summing up the system accuracies of the pressure transducer, torque meter and electromagnetic flowmeter (as shown in Table 1). The results in Table 6 indicate that the validation is successfully achieved using the level of validation uncertainty U within 1%D.In conclusions, the simulation results are convincing by three sets of successively refined meshes. Considering the precision and the cost of calculation, Scheme 2 with 1,521,785 elements was employed in the following calculation. Research on the characteristics of the transient pressure fluctuation Comparisons between experimental data and numerical results The available net positive suction head (NPSH ) that corresponds to a 3% drop in the head is the critical cavitation point of the pump, namely NPSH . The comparison of the cavitation performance and the pressure 3% fluctuation at the outlet of the volute at NPSH is shown in Figure 6, in which the trend of the cavitation 3% performance curve and the amplitude of the dominant frequency exhibit a good correspondence. As the value of NPSH decreases from 8 m to 1 m, calculation errors are lower than 2%; besides, the difference of NPSH a 3% between experiment and simulation is 0.133 m (as shown in Figure 6(a)). As observed from Figure 6(b), the abscissa represents the ratio of a certain frequency to the rotation frequency, f , of the impeller, whereas 4f i i indicates the blade frequency; the ordinate indicates the pressure fluctuation coefficient, c =p/0.5q u , where p l in u illustrates the inlet velocity. The calculation error at the dominant frequency is lower than 2.5%, which is in 42,43 regarded as a fairly good result for the pulsation pressure prediction. Meanwhile, the simulation and the experiment both capture an obvious peak in the vicinity of 1/6f . It proves that excitation sources originate from internal perturbation, which is what the research wants to reveal. Although there is a considerable deviation in high frequency regions owing to the regardless of the environment disturbance in the simulation, it is not the research focus. According to the comparisons between experimental data and numerical results, the accuracy of the numerical method can be convincing. Analyses on the unsteady pressure characteristics under different cavitation degrees The drop in head can reflect the degree of cavitation development, so three points in Figure 6(a) with approximate head drop of 1%, 2% and 3%, respectively (NPSH ¼ 2.986 m, NPSH =1.584 m and NPSH =1.186 m), 1% 2% 3% are selected as the typical ones to describe the influence of cavitation development on pressure fluctuation. For simplicity, the frequency values were normalized by the rotation frequency, f , which was 48.333 Hz. And the harmonic frequencies were represented as 2f,3f , etc. i i Figures 7 and 8 show frequency domain characteristics and maximum amplitudes of pressure fluctuation at the monitoring points under different cavitation degrees. It is obvious to find that there are some interlinked phe- nomena at a majority of monitoring points in diagrams below, for example, the energy of pulsation pressure is concentrated mainly in low-frequency regions, the dominant frequency is the rotation frequency f or its harmonic frequencies, the maximum amplitudes of pressure fluctuation gradually increase from inlet to outlet and from the suction side to the pressure side, whereas maximum amplitudes gradually decrease with the evolution of cavitation. However, there are some special phenomena due to the complex internal flow field. With the Wang et al. 535 Figure 6. Comparisons between experimental data and numerical results: (a) cavitation performance; (b) pressure fluctuation at the outlet of the volute at NPSH . 3% development of cavitation, there are more fluctuation components appearing in the region below f and the amplitude gradually increases at the monitoring point S , which lies in the vicinity of the blade leading edge. As observed in Figure 7(c), when critical cavitation occurs, 1/6f becomes the dominant frequency and the ampli- tude increases at S . It makes S the only point on the suction surface where the amplitude increases with the 2 2 decrease of NPSH . It is no doubt that the occurrence of cavitation will reduce the supercharging capacity of the impeller, but there are two points (P and P ) on the pressure surface that amplitudes increase like S (see in 4 5 2 Figure 8(c)). At NPSH =1.186 m, the amplitude of pressure fluctuation at the monitoring S becomes 0, which a 1 means that point lies in the cavity and the pressure remains constant. Figure 9(a) demonstrates the distribution of the vortex near to S at NPSH =1.186 m at three typical instants, 2 a where t in the figure is the beginning of the third period. In the six periods, the vortex affecting S undergo a quasi- periodic variation from downstream to adjacent S and finally moves downstream, while the quasi-periodic variation of the vortex corresponds to about 1/6f . Validation of the vortex frequency at S is illustrated in i 2 Figure 9(b). It can be seen that the frequency domain characteristics of the vortex are similar to pressure fluc- tuation except for the difference in magnitude. When NPSH is lower than 1.584 m, the peak of vorticity is at 1/6f a i approximately and gradually increases. Thus, it is believed that the special phenomena mentioned above in the pressure spectrum are inextricably linked to the change of the vortex field. Consequently, it is appropriate to introduce the vortex transport equation to get a better understanding of mechanisms of the pressure fluctuation– vortex interaction in the ultra-low specific-speed centrifugal pump. 536 Journal of Low Frequency Noise, Vibration and Active Control 38(2) Figure 7. Frequency domain charts of different monitoring points under different cavitation conditions. (a) Frequency spectrogram on the suction surface under NPSH ¼2.986 m. (b) Frequency spectrogram on the suction surface under NPSH ¼1.548 m. (c) a a Frequency spectrogram on the suction surface under NPSH ¼1.186 m. (d) Frequency spectrogram on the middle surface under NPSH ¼2.986 m. (e) Frequency spectrogram on the middle surface under NPSH ¼1.548 m. (f) Frequency spectrogram on the middle a a surface under NPSH ¼1.186 m. (g) Frequency spectrogram on the pressure surface under NPSH ¼2.986 m. (h) Frequency spec- a a trogram on the pressure surface under NPSH ¼1.548 m. (i) Frequency spectrogram on the pressure surface under NPSH ¼1.186 m. a a The relative vortex transport equation and the pressure fluctuation–vortex interaction As for vortex transport equation in the rotational frame, considering the Coriolis force, the equation can be written as follows DX rq rp ðÞ ¼ðÞ X r W X r W þ r ðÞ X W þ r X (8) r r r r Dt q Here, X is the vectorial relative vorticity, W is the vectorial relative velocity and is the kinematic viscosity. The first two terms on the right-hand side (RHS) represent the relative vortex stretching, the relative vortex dilation (expansion or contraction). And the last three terms originate from the baroclinic torque, the Coriolis force and the viscous diffusion. Since the impeller revolves around Z axis, the results are mainly discussed in the XY plane. The relative vortex transport equation is given as DX rq rp r;z ¼ þ r X þð½ W rÞX (9) r r z z Dt q The first term on the RHS is the baroclinic torque, which acts whenever pressure and density gradients are not aligned. The next term is named as the viscous diffusion, which leads to the diffusion in vorticity. As for the last term on the RHS, it is originated from the vorticity gradients, which called the vortex convection. Wang et al. 537 Figure 8. Maximum amplitudes of pressure fluctuation at different monitoring points: (a) the suction surfaces, (b) the middle surfaces and (c) the pressure surfaces. Figure 9. (a) Streamline in the passages of the impeller at NPSH ¼1.186 m at three typical instants. (b) Monitoring of the vorticity at S . 2 538 Journal of Low Frequency Noise, Vibration and Active Control 38(2) Figure 10. Streamlines in passages of the impeller with the evolution of cavitation at the beginning of the last period. (a) NPSH ¼2.986 m, (b) NPSH ¼1.548 m, (c) NPSH ¼1.186 m, (d) NPSH ¼2.986 m, (e) NPSH ¼1.548 m and (f) NPSH ¼1.186 m. a a a a a a Streamlined profile is a simple method for vortex identification and the spiraling or closed streamlines are intuitional to depict the pattern of a vortex. Figure 10 shows streamlines in passages of the impeller with the evolution of cavitation at the beginning of the last period, while magnitude of radial velocity v implies the trend of vortex motion. At NPSH =2.986 m in Figure 10(a), vortices are relatively symmetrical in four channels. During this process, all vortices are generated by flow separation when the boundary layer travels far enough against an adverse pressure gradient that the speed of the boundary layer relative to the object falls almost to zero, and hence the rotation direction of them is just opposite to that of the impeller. And at NPSH =1.548 m, there are certain smaller-scale vortices on the suction surface of the blade. Comparing the position of the separated vortices, the emerging vortices are located upstream of the flow channel. From Figure 10(e), it can be observed that the emerging vortices that have the same rotating direction as the impeller are closely associated with the cavitation, which are called cavitating vortices. When compared with the condition of non-cavitation, the radial velocity increases obviously. Then at NPSH =1.186 m, namely the critical cavitation point, the vapor region has devel- oped from the suction surface of the blade to the pressure surface. More and larger-scale cavitating vortices are formed at the upstream of the channels affected by the expansion of the cavitation region. From Figure 10(f), cavitating vortices makes the channels narrower and the radial velocity higher, which makes the separation vortices move downstream. Therefore, the process greatly promotes the vorticity transport in space. Figure 11 shows the distribution for all terms with the evolution of cavitation on the RHS of equation (9). The baroclinic torque is significant for cavitating vortices generation during the evolution of cavitation because pres- sure gradients are not in parallel with density gradients after cavitation, whereas it is zero under non-cavitation conditions owing to the density remain constant. In Figure 11(a), when NPSH drops to 1.186 m, S is located a 2 near the interface between vapor and liquid, whereas P and P lie away from the cavity region. Therefore, the 4 5 baroclinic torque is an engine for the obvious growth of the pressure fluctuation amplitude at S , yet it does not affect P and P . As is shown in Figure 11(b), the effect of viscous diffusion term is neglected when compared with 4 5 the other two terms. According to the fundamentals of vorticity dynamics illustrated in Wu et al., vorticity diffusion flux will be weak when the Reynolds number is much greater than 1. Due to the Reynolds number exceeds 10 in this research, it is appropriate to neglect the process of viscous diffusion. Although the baroclinic Wang et al. 539 Figure 11. Comparison for all terms on the RHF of equation (2) (first column: NPSH ¼2.986 m; second column: NPSH ¼1.584 m; a a third column: NPSH ¼1.186 m). (a) Comparison of baroclinic torque contours. (b) Comparison of viscous diffusion contours. (c) Comparison of vortex convection contours. torque is crucial, the main contributor to the change of fluctuation amplitude is the vortex convection. As is shown in Figure 11(c), although the value of the vortex convection at S is high, there is no significant change with cavitation development. In contrast, P and P have a rapid increase because of the downstream movement of the 4 5 vortices (see Figure 10). Besides, the baroclinic torque and the vortex convection that mainly cause the change of the relative vorticity value both decrease or keep unchanged with the deterioration of cavitation at the remaining monitoring points, which correspond to the principle of pressure fluctuation. Based on above description and discussion, it can be concluded that the vortex transport equation can intuitionally illustrate the fluctuation amplitude variation at S ,P /P other monitoring points. 2 4 5 Meanwhile, it can be concluded that the change of pressure fluctuation amplitude at P /P remains far from 4 5 clear when the cavitation effect is merely concerned. Whereas the cavitation effect as well as the vortex convec- tion/diffusion via the relative vortex transport equation can blend together and clarify the abnormal phenomena in frequency spectrum intuitionally. It is not difficult to find that there is an inherent relation between pressure fluctuation and vortex field. Pressure fluctuation is the acoustic expression of vortex field, whereas the change of vortex field becomes the excitation source of pressure fluctuation. Since the typical characteristics of pressure fluctuation can be analyzed intuitively 540 Journal of Low Frequency Noise, Vibration and Active Control 38(2) by the vortex transport equation, even with different pumps, the study of the pressure fluctuation–vortex inter- action will enrich the research of the instability in the dynamic machinery. Conclusions In this paper, the unsteady flow in an ultra-low specific-speed centrifugal pump was simulated by applying the RNG k–e turbulence model and the transport equation-based cavitation model to provide a better insight in the pressure fluctuation–vortex interaction. The following conclusions can be drawn based on the present investigation. The uncertainty of the pump head and efficiency from the numerical simulations with three sets of successively refined mesh were verified and validated by a level of 1% of the experimental results. Then, using the validated numerical framework, the calculation of the cavitation performance and pressure fluctuation at the outlet of the volute at NPSH agrees reasonably well with the experimental results. 3% Vortex development induces flow unsteadiness, which is manifested in some transient phenomena that happen in pressure spectrum. The phenomena are believed to have inextricable links with the vorticity field, including abnormal increasing amplitudes at the monitoring points S on the suction surface and P /P on the pressure 2 4 5 surface in the case of lower pressurization capacity of impeller after cavitation. By the contribution of baroclinic torque, viscous diffusion and vortex convection terms, further analyses based on the relative vortex transport equation illustrate that vorticity diffusion is weak when the Reynolds number is much greater than 1, which is neglected when compared with the other two terms. Pressure fluctuation amplitude enlarges at S due to the baroclinic torque in cavitation regions, which makes S the only point on the suction 2 2 surface where the amplitude increases with the development of cavitation, whereas the abnormal increase of pressure fluctuation after cavitation at P and P results from the vortex convection during vortices moving 4 5 downstream with the decrease of NPSH . When it concerns merely the cavitation effect, the abnormal increasing of pressure fluctuation amplitude at P /P cannot be explained reasonably. However, through the relative vortex transport equation, the cavitation 4 5 effect as well as the vortex convection/diffusion can blend together and clarify the abnormal phenomena in frequency spectrum intuitionally. Briefly, it can be said that pressure fluctuation is the acoustic expression of vortex field, whereas the change of vortex field becomes the excitation source of pressure fluctuation. Finally, since the typical characteristics of pressure fluctuation can be analyzed intuitively by the relative vortex transport equation, the study of the pressure fluctuation–vortex interaction for the first time is supposed to pave the way for further research of the instability in the ultra-low specific-speed centrifugal pump. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/ or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant Nos. 51209217 and 51876220). ORCID iD Cong Wang http://orcid.org/0000-0002-4781-5165 References 1. Feng J, Wu H, Luo X, et al. Measurements of internal flow in centrifugal pump at part-load conditions based on LDV. J Drain Irrig Mach Eng 2013; 31: 109–112 þ 122. 2. Zhang N, Yang M, Gao B, et al. Investigation of rotor-stator interaction and flow unsteadiness in a low specific speed centrifugal pump. STROJ VESTN-J MECH E 2016; 62: 21–31. 3. Zhu B and Chen HX. Cavitating suppression of low specific speed centrifugal pump with gap drainage blades. J Hydrodyn Ser B 2012; 24: 729–736. 4. Lu G, Zuo Z, Sun Y, et al. Experimental evidence of cavitation influences on the positive slope on the pump performance curve of a low specific speed model pump-turbine. Renewable Energy 2017; 113: 1539–1550. Wang et al. 541 5. Ji P, Wang WJ and Yuan SQ. Statistical analysis of pressure fluctuations during unsteady flow for low-specific-speed centrifugal pumps. J Cent South Univ 2014; 21: 1017–1024. 6. Yang X, Wu C, Wen H, et al. Numerical simulation and experimental research on the aerodynamic performance of large marine axial flow fan with a perforated blade. J Low Freq Noise Vib Active Control 2018; 37: 410–421. 7. Seol H. Time domain method for the prediction of pressure fluctuation induced by propeller sheet cavitation: Numerical simulations and experimental validation. Ocean Eng 2013; 72: 287–296. 8. Chen G, Wang G, Hu C, et al. Combined experimental and computational investigation of cavitation evolution and excited pressure fluctuation in a convergent-divergent channel. Int J Multiphase Flow 2015; 72: 133–140. 9. Ji B, Luo X, Wu Y, et al. Numerical investigation of three-dimensional cavitation evolution and excited pressure fluctua- tions around a twisted hydrofoil. J Mech Sci Technol 2014; 28: 2659–2668. 10. Jia XQ, Cui BL, Zhu ZC, et al. Experimental investigation of pressure fluctuations on inner wall of a centrifugal pump. Int J Turbo Jet Engines 2017. DOI: 10.1515/tjj-2016-0078 11. Liu QZ, Yang K, Li DY, et al. Research of fluid-induced pressure fluctuation due to impeller-volute interaction in a centrifugal pump. In: IOP Conference series: materials science and engineering, 2013; 52: 022026. 12. Tan L, Zhu BS, Cao SL, et al. Numerical simulation of unsteady cavitation flow in a centrifugal pump at off-design conditions. J Mech Eng Sci 2013; 228: 1994–2006. 13. Zhang Y, Chen T, Li J, et al. Experimental study of load variations on pressure fluctuations in a prototype reversible pump turbine in generating mode. J Fluids Eng 2017; 139: 074501. 14. Zhang Y, Zhang J and Hou H. Study on pressure pulsation in the volute of a centrifugal pump by large eddy simulation. In: ASME 2014 international mechanical engineering congress and exposition 2014; 7: 1–7. 15. Hou HC, Zhang YX, Li ZL, et al. A CFD study of IGV vane number on hydraulic characteristics and pressure pulsation of an IS centrifugal pump. J Vibroeng 2017; 19: 563–576. 16. Arndt REA. Cavitation in vortical flows. Annu Rev Fluid Mech 2002; 34: 143–175. 17. Le Q, Franc JP and Michel JM. Partial cavities: global behavior and mean pressure distribution. J Fluids Eng 1993; 115: 243–248. 18. Kimura T and Yoshida Y. Numerical simulation for vortex structure in a turbopump inducer: close relationship with appearance of cavitation instabilities. J Fluids Eng 2008; 130: 051104. 19. Ji B, Luo X, Arndt REA, et al. Numerical simulation of three dimensional cavitation shedding dynamics with special emphasis on cavitation–vortex interaction. Ocean Eng 2014; 87: 64–77. 20. Huang B, Zhao Y and Wang G. Large eddy simulation of turbulent vortex-cavitation interactions in transient sheet/cloud cavitating flows. Comput Fluids 2014; 92: 113–124. 21. Huang R, Ji B, Luo X, et al. Numerical investigation of cavitation-vortex interaction in a mixed-flow waterjet pump. J Mech Sci Technol 2015; 29: 3707–3716. 22. Anup KC, Lee YH and Thapa B. CFD study on prediction of vortex shedding in draft tube of Francis turbine and vortex control techniques. Renewable Energy 2016; 86: 1406–1421. 23. Gao C, Sun G, Dong R, et al. Characterizing the dynamic property of the vortex tail in a gas cyclone by wall pressure measurements. Fuel Process Technol 2010; 91: 921–926. 24. Gu X, Song J and Wei Y. Experimental study of pressure fluctuation in a gas-solid cyclone separator. Powder Technol 2016; 299: 217–225. 25. Luo X, Yu A, Ji B, et al. Unsteady vortical flow simulation in a Francis turbine with special emphasis on vortex rope behavior and pressure fluctuation alleviation. Proc Inst Mech Eng Part A J Power Energy 2017; 231: 215–226. 26. Gao B, Zhang N, Li Z, et al. Influence of the blade trailing edge profile on the performance and unsteady pressure pulsation in a low specific speed centrifugal pump. J Fluids Eng 2016; 138: 051106-01–051106-10. 27. Branlard E. Wind turbine aerodynamics and vorticity-based methods. Switzerland: Springer International Publishing, 2017, pp.40–44. 28. Kumar DS, Murugesan K and Thomas HR. Numerical simulation of double diffusive mixed convection in a lid-driven square cavity using velocity-vorticity formulation. Numer Heat Transfer 2008; 54: 837–865. 29. Dittakavi N, Chunekar A and Frankel S. Large eddy simulation of turbulent-cavitation interactions in a Venturi nozzle. J Fluids Eng 2010; 132: 121301–121311. 30. Li D, Wang H, Qin Y, et al. Entropy production analysis of hysteresis characteristic of a pump-turbine model. Energy Convers Manage 2017; 149: 175–191. 31. Salehi S, Raisee M, Cervantes MJ, et al. On the flow field and performance of a centrifugal pump under operational and geometrical uncertainties. Appl Math Modelling 2018; 61: 540–560. 32. Wang C, Shi W, Wang X, et al. Optimal design of multistage centrifugal pump based on the combined energy loss model and computational fluid dynamics. Appl Energy 2017; 187: 10–26. 33. Zhu B and Chen HX. Cavitating suppression of low specific speed centrifugal pump with gap drainage blades. J Hydrodynam 2012; 24: 729–736. 34. Cai JC, Qi DT, Lu FA, et al. Study of the tonal casing noise of a centrifugal fan at the blade passing frequency. Part I. Aeroacoustics. J Low Freq Noise Vib Active Control 2010; 29: 253–266. 542 Journal of Low Frequency Noise, Vibration and Active Control 38(2) 35. Lu FA, Wang XJ, Qi DT, et al. Study of the tonal casing noise of a centrifugal fan at the blade passing frequency. Part II. Vibroacoustics. J Low Freq Noise Vib Active Control 2010; 30: 89–105. 36. Long Y, Wang D, Yin J, et al. Numerical investigation on the unsteady characteristics of reactor coolant pumps with non- uniform inflow. Nucl Eng Des 2017; 320: 65–76. 37. Pei J, Dohmen HJ, Yuan SQ, et al. Investigation of unsteady flow-induced impeller oscillations of a single-blade pump under off-design conditions. J Fluids Struct 2012; 35: 89–104. 38. Su X, Huang S, Zhang X, et al. Numerical research on unsteady flow rate characteristics of pump as turbine. Renewable Energy 2016; 94: 488–495. 39. Pei J, Yuan SQ, Li XJ, et al. Numerical prediction of 3-D periodic flow unsteadiness in a centrifugal pump under part-load condition. J Hydrodynam Ser B 2014; 26: 257–263. 40. ITTC. Guide to the expression of uncertainty in experimental hydrodynamics (7.5-02-01-01). ITTC-Recommended Procedures and Guidelines, 2008. 41. Qiu JT, Yang CJ, Dong XQ, et al. Numerical simulation and uncertainty analysis of an axial-flow waterjet pump. J Mar Sci Eng 2018; 6: 71. 42. Barrio R, Ferna´ ndez J, Blanco E, et al. Estimation of radial load in centrifugal pumps using computational fluid dynamics. Eur J Mech 2011; 30: 316–324. 43. Rau B, Eduardo B, et al. The effect of impeller cutback on the fluid-dynamic pulsations and load at the blade-passing frequency in a centrifugal pump. J Fluids Eng 2008; 130: 1349–1357. 44. Foeth EJ. The structure of three-dimensional sheet cavitation. Delft, Holland: Delft University of Technology, 2008, pp.47–56. 45. Zhang Y, Liu K, Xian H, et al. A review of methods for vortex identification in hydroturbines. Renewable Sustainable Energy Rev 2017; 81: 1269–1285. 46. Wu JZ, Ma HY and Zhou MD. Vorticity and vortex dynamics. Berlin, Germany: Springer Berlin Heidelberg, 2006, pp.138–144. Appendix Notation c pressure fluctuation coefficient C correction factor D experimental data E comparison error f rotation frequency, Hz H pump head, m m mesh elements n rotational speed, r/min n dimensionless specific speed p pressure, Pa p order of accuracy P /P the monitoring points near to the trailing edge on the pressure surface 4 5 Q design discharge, m /h r mesh refinement ratio R mesh convergence ratio S /S maximum and minimum values of results in the last period U L S the monitoring point near to the leading edge on the suction surface T rotation period, s u velocity, m/s u inlet velocity, m/s in U uncertainty W vectorial relative velocity, m/s Z number of blades a volume fraction d Kronecker delta function ij e numerical difference between meshes -1 X vectorial relative vorticity, s r Wang et al. 543 g hydraulic efficiency l mixture viscosity, Pas l turbulent viscosity, Pas kinematic viscosity, m /s q mixture density, kg/m -1 x angular velocity, s Subscripts and superscripts i different mesh items SN numerical data I the uncertainties resulted from iteration M the uncertainties resulted from mesh density T the uncertainties resulted from time-step size P the uncertainties yielded from other parameters F–M difference between fine mesh and medium one M–C difference between medium mesh and coarse one D experimental data V validation value
"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE
Published: Dec 10, 2018
Keywords: Unsteady cavitation flow; an ultra-low specific-speed centrifugal pump; pressure fluctuation; the vortex transport equation; numerical simulation
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