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Review Article

A Review of Available Methods for the Assessment of Fluid Added Mass, Damping, and Stiffness With an Emphasis on Hydraulic Turbines

[+] Author and Article Information
Arash Soltani Dehkharqani

Division of Fluid and Experimental Mechanics,
Luleå University of Technology,
Luleå, SE-971 87, Sweden
e-mail: arasol@ltu.se

Jan-Olov Aidanpää

Division of Product and
Production Development,
Luleå University of Technology,
Luleå, SE-971 87, Sweden
e-mail: Jan-Olov.Aidanpaa@ltu.se

Fredrik Engström

Division of Fluid and Experimental Mechanics,
Luleå University of Technology,
Luleå, SE-971 87, Sweden
e-mail: fredrik.1.engstrom@ltu.se

Michel J. Cervantes

Division of Fluid and Experimental Mechanics,
Luleå University of Technology,
Luleå, SE-971 87, Sweden
e-mail: Michel.Cervantes@ltu.se

1Corresponding author.

Manuscript received April 25, 2018; final manuscript received December 13, 2018; published online January 8, 2019. Editor: Harry Dankowicz.

Appl. Mech. Rev 70(5), 050801 (Jan 08, 2019) (20 pages) Paper No: AMR-18-1052; doi: 10.1115/1.4042279 History: Received April 25, 2018; Revised December 13, 2018

Fluid added mass, damping, and stiffness are highly relevant parameters to consider when evaluating the dynamic response of a submerged structure in a fluid. The prediction of these parameters for hydraulic turbines has been approached relatively recently. Complex fluid-structure analyses including three-dimensional flow and the need for experiments during operation are the main challenges for the numerical and experimental approaches, respectively. The main objective of this review is to address the impact of different parameters, for example, flow velocity, cavitation, nearby solid structure, and rotational speed on the fluid added mass and damping of Kaplan/Propeller and Francis turbine runners. The fluid added stiffness is also discussed in the last section of the paper. Although studies related to hydraulic turbines are the main objective of this paper, the literature on hydrofoils is also taken into consideration to provide valuable information on topics such as individual runner blades. In this literature survey, the analytical, numerical, and experimental approaches used to determine fluid added parameters are discussed, and the pros and the cons of each method are addressed.

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Wu, Y. , Li, S. , Liu, S. , Dou, H. , and Qian, Z. , 2013, Vibration of Hydraulic Machinery, Springer, London.
Hengstler, J. A. , 2013, “ Influence of the Fluid-Structure Interaction on the Vibrations of Structures,” Ph.D. thesis, ETH Zurich, Zürich, Switzerland. https://www.research-collection.ethz.ch/bitstream/handle/20.500.11850/76519/eth-7837-01.pdf
Presas, A. , Egusquiza, E. , Valero, C. , Valentin, D. , and Seidel, U. , 2014, “ Feasibility of Using PZT Actuators to Study the Dynamic Behavior of a Rotating Disk Due to Rotor-Stator Interaction,” Sensors, 14(7), pp. 11919–11942. [CrossRef] [PubMed]
Presas, A. , Valentin, D. , Egusquiza, E. , Valero, C. , and Seidel, U. , 2015, “ Influence of the Rotation on the Natural Frequencies of a Submerged-Confined Disk in Water,” J. Sound Vib., 337, pp. 161–180. [CrossRef]
Mehdigholi, H. , 1991, “ Forced Vibration of Rotating Discs and Interaction with Non-Rotating Structures,” Ph.D. thesis, University of London, London.
Kubota, Y. , and Ohashi, H. , 1991, “ A Study on the Natural Frequencies of Hydraulic Pumps,” First ASME Joint International Conference on Nuclear Engineering, Tokyo, Japan, Nov. 4–7, pp. 589–593.
Valentín, D. , Presas, A. , Egusquiza, E. , and Valero, C. , 2016, “ On the Capability of Structural–Acoustical Fluid–Structure Interaction Simulations to Predict Natural Frequencies of Rotating Disklike Structures Submerged in a Heavy Fluid,” ASME J. Vib. Acoust., 138(3), p. 034502. [CrossRef]
Blevins, R. D. , 1990, Flow-Induced Vibration, Krieger Publication Company, Malabar, FL.
Yamamoto, T. , 1983, “ On the Response of a Coulomb‐Damped Poroelastic Bed to Water Waves,” Mar. Georesour. Geotechnol., 5(2), pp. 93–130. [CrossRef]
Kareem, A. , and Gurley, K. , 1996, “ Damping in Structures: Its Evaluation and Treatment of Uncertainty,” J. Wind Eng. Ind. Aerodyn., 59(2–3), pp. 131–157. [CrossRef]
Isomura, K. , and Giles, M. B. , 1998, “ A Numerical Study of Flutter in a Transonic Fan,” ASME J. Turbomach., 120(3), pp. 500–507. [CrossRef]
Kammerer, A. , and Abhari, R. S. , 2009, “ Experimental Study on Impeller Blade Vibration During Resonance—Part II: Blade Damping,” ASME J. Eng. Gas Turbines Power, 131(2), p. 022508. [CrossRef]
Grüber, B. , and Carstens, V. , 2000, “ The Impact of Viscous Effects on the Aerodynamic Damping of Vibrating Transonic Compressor Blades—A Numerical Study,” ASME J. Turbomach., 123(2), pp. 409–417. [CrossRef]
Kaminer, A. A. , and Kavitskii, B. M. , 1976, “ Experimental Investigation of Hydrodynamic Damping During Bending Oscillations of Blade Profiles in Water Flow,” Strength Mater., 8(1), pp. 25–27. [CrossRef]
Liaghat, T. , Guibault, F. , Allenbach, L. , and Nennemann, B. , 2014, “ Two-Way Fluid-Structure Coupling in Vibration and Damping Analysis of an Oscillating Hydrofoil,” ASME Paper No. IMECE2014-38441.
Kielb, J. J. , and Abhari, R. S. , 2003, “ Experimental Study of Aerodynamic and Structural Damping in a Full-Scale Rotating Turbine,” ASME J. Eng. Gas Turbines Power, 125(1), pp. 102–112. [CrossRef]
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Zobeiri, A. , 2012, “ Effect of Hydrofoil Trailing Edge Geometry on the Wake Dynamics,” Ph.D. thesis, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland. https://infoscience.epfl.ch/record/168992/files/EPFL_TH5218.pdf

Figures

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Fig. 1

The hydrofoil (H1) and installed instruments. (Reproduced with permission from Seeley et al. [23]. Copyright 2013 by GE).

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Fig. 2

Meshed fluid and structure domain and specified boundary conditions. (Reproduced with permission from de Souza Braga et al. [28]. Copyright 2013 by ABCM).

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Fig. 3

The FEM model of the Francis turbine with and without clearance. (Reproduced with permission from Zhongyu and Zhengwei [68]. Copyright 2016 by Wang Zhengwei).

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Fig. 4

Natural frequencies of the Kaplan runner in air and water [28]. The hollow symbols correspond to natural frequencies in water obtained from simulation. The square and diamond solid symbols correspond to natural frequencies in air obtained from experiment and simulation, respectively. (Reproduced with permission from de Souza Braga et al. [28]. Copyright 2013 by ABCM).

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Fig. 7

Different cavity sizes at a flow velocity of 14 m/s, different sigma values, and incidence angles of 1 deg (top) and 2 deg (bottom) [25]. l and c are the cavity length and profile chord length, respectively. (Reproduced with permission from De La Torre et al. [25]. Copyright 2013 by Elsevier).

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Fig. 8

Hydrofoil mode shapes at different flow conditions with and without cavitation [69]. CSR=l/(2c); CSR is the ratio of the area of the hydrofoil surface covered by the sheet cavity, l is the cavity length, and c is the profile chord length. (Reproduced with permission from Liu et al. [69]. Copyright 2017 by ASME).

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Fig. 5

The natural frequency of the first bending mode for three hydrofoils (H0, H1, and H3) with respect to flowing water velocity. (Reproduced with permission from Seeley et al. [21]. Copyright 2012 by IOP Publishing).

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Fig. 6

Added mass coefficient variation with respect to σ/2α for f2 and f3 at the incidence angles of 1 deg and 2 deg. Cm=(fair/ffluid)2−1 is the added mass coefficient; σ=(Pr−Pv)/(1/2ρV2) is the sigma value, where Pr, Pv, and V represent the pressure at the inlet section, the vapor pressure, and the free stream velocity at the inlet, respectively. α=yυ/(yυ+yl) is the void fraction, where yυ and yl are the volume of the cavity in the gas and liquid phases, respectively. (Reproduced with permission from De La Torre et al. [25]. Copyright 2013 by Elsevier).

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Fig. 9

Schematic of a hydrofoil cross section, where ld is the starting point of cavitation from the LE. The chord of the hydrofoil is 100 mm. (Reproduced with permission from Liu et al. [69]. Copyright 2017 by ASME).

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Fig. 10

Added mass coefficient variation of the first three modes for different starting points. (Reproduced with permission from Liu et al. [69]. Copyright 2017 by ASME).

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Fig. 11

Added mass coefficient for the first bending mode versus submergence level with TE orientation. (Reproduced with permission from De La Torre et al. [26]. Copyright 2013 by Elsevier).

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Fig. 14

Experimental and numerical added mass coefficients versus different gap distances for the first bending mode (a), first torsion mode (b), and second bending mode (c). (Reproduced with permission from De La Torre et al. [26]. Copyright 2013 by Elsevier).

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Fig. 18

Natural frequencies and corresponding phases of the disk in air, still water, and rotating in air and water [104]

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Fig. 15

The natural frequency ratios of different modes for different gap sizes. fw and fa are the natural frequencies in water and air, respectively. From the bottom, the lines correspond to second, third, fourth, and fifth bending modes and the radial mode of the shaft. (Reproduced with permission from Valentín et al. [98]. Copyright 2014 by Elsevier).

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Fig. 16

Added polar inertia as a function of the perturbation frequency obtained with zero and nonzero stiffness assumption [34]

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Fig. 17

Phase and amplitude diagram of the vibrating circular plate for the second mode. (Reproduced with permission from Hengstler [103]. Copyright 2013 by Johannes A. N. Hengstler).

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Fig. 12

Natural frequencies of a circular plate function of the water height (H1 corresponds to the water height over the plate). From the bottom of the figure, the lines correspond to nodal diameters of 0, 1, 2, and 3, respectively. (Reproduced with permission from Askari et al. [97]. Copyright 2013 by Elsevier).

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Fig. 13

Effect of a lateral gap on different modes for a circular plate. The curves with the solid and hollow symbols correspond to zero and one nodal diameter, respectively. The square, circle, upward-pointing triangle, and downward-pointing triangle correspond to one, two, three, and four nodal circles, respectively. (Reproduced with permission from Askari et al. [97]. Copyright 2013 by Elsevier).

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Fig. 19

The damping ratio as a function of flow velocity: (a) experimental results for the first bending mode for H0, H1, and H3 hydrofoils (Reproduced with permission from Seeley et al. [23]. Copyright 2013 by GE), (b) numerical and experimental results for H0 hydrofoil [44], and (c) results obtained by modal work and one-DOF methods for H0 hydrofoil [44]. (Reproduced with permission from Monette et al. [44]. Copyright 2014 by IOP Publishing).

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Fig. 20

Dimensionless added damping as a function of reduced velocity [32]. μ*=μf/((1/4)ρCrefAL2); μ* is the dimensionless added damping, μf is the fluid damping, ρ is the fluid density, Cref is the upstream velocity, L is the chord length, and A=L×e is the cross-sectional area, where e is the numerical domain span. (Reproduced with permission from Münch et al. [32]. Copyright 2010 by Elsevier).

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Fig. 22

Dimensionless fluid added stiffness as a function of reduced velocity [32]. k*=kf/((1/2)ρCrefAL2); k* is the dimensionless added stiffness, kf is the fluid added stiffness, ρ is the fluid density, Cref is the upstream velocity, L is the chord length, and A=L×e is the cross-sectional area, where e is the numerical domain span. (Reproduced with permission from Münch et al. [32]. Copyright 2010 by Elsevier).

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Fig. 21

The damping factor as a function of reduced velocity for Donaldson and blunt TEs. (a) First bending mode and (b) first torsion mode. (Reproduced with permission from Yao et al. [20]. Copyright 2014 by Elsevier).

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