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Research Papers: Flows in Complex Systems

Numerical Study of the Winter-Kennedy Method—A Sensitivity Analysis

[+] Author and Article Information
Binaya Baidar

Department of Engineering
Sciences and Mathematics,
Luleå University of Technology,
Luleå 971 87, Sweden
e-mail: binaya.baidar@ltu.se

Jonathan Nicolle

Mécanique, métallurgie et hydro-éolien,
Instit de recherche d'Hydro-Québec,
Varennes, QC J3X 1S1, Canada
e-mail: nicolle.jonathan@ireq.ca

Chirag Trivedi

Department of Energy and Process Engineering,
Norwegian University of Science
and Technology,
Trondheim 7491, Norway
e-mail: chirag.trivedi@ntnu.no

Michel J. Cervantes

Professor
Department of Engineering
Sciences and Mathematics,
Luleå University of Technology,
Luleå 971 87, Sweden;
Department of Energy and Process Engineering,
Norwegian University of Science
and Technology,
Trondheim 7491, Norway
e-mail: michel.cervantes@ltu.se

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received May 19, 2017; final manuscript received October 27, 2017; published online January 9, 2018. Assoc. Editor: Oleg Schilling.

J. Fluids Eng 140(5), 051103 (Jan 09, 2018) (11 pages) Paper No: FE-17-1285; doi: 10.1115/1.4038662 History: Received May 19, 2017; Revised October 27, 2017

The Winter-Kennedy (WK) method is commonly used in relative discharge measurement and to quantify efficiency step-up in hydropower refurbishment projects. The method utilizes the differential pressure between two taps located at a radial section of a spiral case, which is related to the discharge with the help of a coefficient and an exponent. Nearly a century old and widely used, the method has shown some discrepancies when the same coefficient is used after a plant upgrade. The reasons are often attributed to local flow changes. To study the change in flow behavior and its impact on the coefficient, a numerical model of a semi-spiral case (SC) has been developed and the numerical results are compared with experimental results. The simulations of the SC have been performed with different inlet boundary conditions. Comparison between an analytical formulation with the computational fluid dynamics (CFD) results shows that the flow inside an SC is highly three-dimensional (3D). The magnitude of the secondary flow is a function of the inlet boundary conditions. The secondary flow affects the vortex flow distribution and hence the coefficients. For the SC considered in this study, the most stable WK configurations are located toward the bottom from θ=30deg to 45deg after the curve of the SC begins, and on the top between two stay vanes.

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Figures

Grahic Jump Location
Fig. 1

Model test rig at the Vattenfall hydraulic machinery laboratory in Älvkarleby, Sweden. The penstock and semi-spiral case are marked with the dashed area in (a). The location of the LDA measurements used for the validation of the CFD results is marked with the dashed line, termed as validation line in (b), taken from Ref. [16].

Grahic Jump Location
Fig. 2

Computational domain showing (a) FP and (b) HP model. Both models contain the semi-spiral case and distributor (stay vanes and guide vanes). The HP model is considered in this study by varying the inlet conditions: (1) the normal or ideal inlet and (2) realistic inlet obtained by simulating the FP model in transient and generating averaged velocity profile at the location of HP inlet marked with the dashed ellipse in FP model.

Grahic Jump Location
Fig. 3

Measurements cross section for θ=30−120deg in (a). WK pressure points and combinations at each cross-sectional plane of spiral case with varying angles in (b).

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Fig. 4

WK coefficient KWK, calculated from Eq. (1) with n = 0.5, for the normal inlet (NI_BC) and realistic inlet (FP_BC) conditions at θ=30deg. Euler solution for the normal inlet case is represented by NI_BC Inlet (Euler setup). The experimental results are from Ref. [13].

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Fig. 5

Tangential and radial velocity profile along the validation line (shown in Fig. 1(b)) for the realistic inlet condition (FP_BC). The experimental results are from Ref. [16]. The error bar in the experimental results is assumed to 1% of the value which is reasonable for LDA measurements.

Grahic Jump Location
Fig. 6

Difference between the analytical discharge calculated from Eq. (8) with respect to that predicted by CFD in the cross sections for θ=30−120deg. NI_BC and FP_BC refer to the normal inlet and the realistic inlet, respectively. The analytically calculated discharge is greater than CFD discharge in all the cross sections as it only accounts for the tangential velocity.

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Fig. 7

Normalized velocity moment ruθ at cross section θ=30 deg and θ=105 deg for the normal inlet (NI_BC) and the realistic inlet (FP_BC) inlet boundary condition showing the closeness to the free vortex theory. ruθ is normalized with ruθ at r*=0.5. The theory requires all the lines to be straight: (a) line P35 (WK3), (b), line P15 (WK1), and (c) line P47 (WK4).

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Fig. 8

Variation of WK coefficient, KWK, calculated from Eq. (1) with n = 0.5 in the circumferential direction from θ=30 deg to 120 deg. NI_BC and FP_BC refer to a normal inlet and realistic inlet, respectively.

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Fig. 9

Variation of the different terms in the radial component of the NS equation for the normal inlet (NI_BC) and the realistic inlet (FP_BC) cases for the WK configuration at θ=30 deg. The legend for all figures is placed at the central plot. r−ri/ro−ri, where ri is the radial coordinate at the inner wall and ro is the radial coordinate at the outer wall of the SC. The NI_BC condition was also simulated with Euler condition and represented as NI_BC (Euler setup) to study the effects of turbulence and viscosity.

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Fig. 10

Tangential velocity contours normalized by the bulk velocity. Velocity vectors are represented by the black arrows. Pressure points for the respective WK considered are shown in the right figure, i.e., in FP_BC inlet.

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Fig. 11

Turbulence kinetic energy at the cross section θ=30 deg for the normal inlet (NI_BC) and the realistic inlet (FP_BC). The plane is just in front of a stay vane. Pressure points and the lines for the respective WK considered are shown in the right figure, i.e., in FP_BC inlet.

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Fig. 12

Total vorticity (s−1) per unit area at the considered cross-sectional planes. The values presented are normalized by the maximum value i.e., at θ=120 deg for NI_BC inlet.

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