Research Papers: Flows in Complex Systems

Computational and Theoretical Analyses of the Precessing Vortex Rope in a Simplified Draft Tube of a Scaled Model of a Francis Turbine

[+] Author and Article Information
Girish K. Rajan

Department of Mechanical Engineering,
Pennsylvania State University,
University Park, PA 16802
e-mail: girish@psu.edu

John M. Cimbala

Department of Mechanical Engineering,
Pennsylvania State University,
University Park, PA 16802
e-mail: jmc6@psu.edu

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received November 2, 2015; final manuscript received September 3, 2016; published online November 3, 2016. Assoc. Editor: Riccardo Mereu.

J. Fluids Eng 139(2), 021102 (Nov 03, 2016) (12 pages) Paper No: FE-15-1793; doi: 10.1115/1.4034693 History: Received November 02, 2015; Revised September 03, 2016

Results on flows in a draft tube of a constant-head, constant-specific speed, model Francis turbine are presented based on computational fluid dynamics (CFD) simulations and theoretical analysis. A three-dimensional, unsteady, Navier–Stokes solver with the detached-eddy simulation (DES) model and the realizable k–ϵ (RKE) model is used to analyze the vortex rope formed at different discharge coefficients. The dominant amplitude of the pressure fluctuations at a fixed point in the draft tube increases by 13 times, and the length of the rope increases by 3.4 times when the operating point of the turbine shifts from a discharge coefficient of 0.37 to 0.34. A perturbation analysis based on a steady, axisymmetric, inviscid, incompressible model for the mean flow is performed to obtain a Sturm–Liouville (SL) system, the solutions of which are oscillatory if the discharge coefficient is greater than 0.3635, and nonoscillatory otherwise.

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Fig. 1

Details of the geometry and mesh. The cross section of the simplified draft tube in the XZ and YZ planes are presented in panels (a) and (b), respectively. Panel (c) presents the meshed three-dimensional geometry, panel (d) shows the details of the mesh in the XZ (or YZ) plane, and panel (e) shows the details of the mesh in the inlet plane.

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Fig. 2

Comparison of CFD results with experimental data [33]. The axial velocity, UZ, and the circumferential velocity, Uθ, are plotted as functions of the radial coordinate, R, along the lines L1 and L2 (see Fig. 1), at two different instants of time. Panels (a) and (b) present the results for ϕ=0.34, while panels (c) and (d) present the results for ϕ=0.37 (BEP). The legend for all cases is included in panel (a).

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Fig. 5

Visualizing the vortex rope. Isosurface contours of (a) pressure, P (−16,600 Pa), (b) λ2 (−1345 1/s2), (c) q (1345 1/s2), (d) real eigen helicity, h (80 1/s), (e) swirling discriminant, Δ (9 × 107 1/s6), and (f) swirling strength, s (35 1/s). All results are at t = 10 s, for ϕ=0.34.

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Fig. 3

Pressure history at a point P (see Fig. 1) on the draft tube wall. Results for ϕ=0.34, ϕ=0.35, ϕ=0.36, and ϕ=0.37 (BEP) are presented in the first, second, third, and fourth rows, respectively. Panels (a)–(d) show the pressure, P, as a function of time, t, for the four different discharge coefficients considered. The horizontal line in (a)–(d) represents the average pressure, Pavg. Panels (e)–(h) show the pressure with the mean subtracted out, as a function of the shifted time, t′=(t−2) s. Panels (i)–(l) show the frequency spectrum (of resolution 0.125 Hz) of the corresponding signals in (e)–(h).

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Fig. 4

The (a) average pressure, Pavg, and (b) the range of pressure fluctuations, ΔP, as functions of the discharge coefficient, ϕ. The pressure data presented are collected at point P (see Fig. 1) on the draft tube wall.

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Fig. 9

The stability characteristics of the SL system at different discharge coefficients

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Fig. 6

The vortex rope visualized via isocontours of q (1345 1/s2) for (a) ϕ=0.34, (b) ϕ=0.35, (c) ϕ=0.36, and (d) ϕ=0.37 (BEP). All results are at t = 10 s.

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Fig. 7

A description of the vortex rope dimensions. The rope depicted here corresponds to a discharge coefficient of ϕ=0.34.

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Fig. 8

The vortex rope dimensions: (a) the rope length in axial direction, l, and (b) the rope half-angle, α, as functions of the discharge coefficient, ϕ




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