Research Papers: Fundamental Issues and Canonical Flows

Optimal Control of Part Load Vortex Rope in Francis Turbines

[+] Author and Article Information
Simon Pasche

Laboratory for Hydraulic Machines,
Department of Mechanical Engineering,
Swiss Federal Institute of Technology (EPFL),
Avenue de Cour 33bis,
Lausanne CH-1007, Switzerland
e-mail: simon.pasche@alumni.epfl.ch

François Avellan

Laboratory for Hydraulic Machines,
Department of Mechanical Engineering,
Swiss Federal Institute of Technology (EPFL),
Avenue de Cour 33bis,
Lausanne CH-1007, Switzerland
e-mail: francois.avellan@epfl.ch

François Gallaire

Laboratory of Fluid Mechanics and Instabilities,
Department of Mechanical Engineering,
Swiss Federal Institute of Technology (EPFL),
Lausanne CH-1015, Switzerland
e-mail: francois.gallaire@epfl.ch

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received June 20, 2018; final manuscript received January 9, 2019; published online February 13, 2019. Assoc. Editor: Matevz Dular.

J. Fluids Eng 141(8), 081203 (Feb 13, 2019) (12 pages) Paper No: FE-18-1425; doi: 10.1115/1.4042560 History: Received June 20, 2018; Revised January 09, 2019

The mitigation of the precessing vortex core developing in the draft tube of Francis turbines operating under part load conditions is crucial to increase the operation flexibility of these hydraulic machines to balance the massive power production of intermittent energy sources. A systematic approach following the optimal control theory is, therefore, presented to control this vortical flow structure. Modal analysis characterizes the part load vortex rope as a self-sustained instability associated with an unstable eigenmode. Based on this physical characteristic, an objective function targeting a zero value of the unstable eigenvalue growth rate is defined and subsequently minimized using an adjoint-based optimization algorithm. We determine an optimal force distribution that successfully quenches the part load vortex rope and sketches the design of a realistic control appendage.

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Grahic Jump Location
Fig. 1

Reduced scale model of the FLINDT Francis turbine (a), pressure sensor location (b), and the two draft tubes considered in the present study: the original elbow draft tube (c) and the axisymmetric Moody type draft tube (d)

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Fig. 2

Schematic of the minimization algorithm applied to control the part load vortex rope

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Fig. 3

Pressure isocontour of the part load vortex rope (CP = −4.1) and instantaneous axial velocity distribution on the turbine meridional plane for the elbow type draft tube (a) and for the Moody type draft tube (b). Time-averaged velocity profiles of the elbow and Moody type draft tube in section S1.3 (c) and section S1.75 (d).

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Fig. 4

Time averaged flow field (a) and turbulent eddy viscosity (b)

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Fig. 5

Peat-to-peak pressure coefficient amplitude (a) and growth rate normalized by the uncontrolled dominant eigenvalue (b) during the minimization procedure. Each symbol corresponds to an iteration.

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Fig. 6

Initial (a), penultimate (b), and ultimate (c) axial flow solutions carried out by minimizing the dominant unstable eigenmode of the mean flow, superimposed with a pressure isocontour (CP = −4.1) making apparent the part load vortex rope

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Fig. 7

Eigenvalue spectrum of azimuthal wavenumber m =1 from the mean turbulent flow of the initial (a), penultimate (b), and ultimate (c) iteration. The symbol (*) represents the direct eigenvalues from Eq. (2), and the symbol (∘) represents the targeted eigenvalue associated with the part load vortex rope, ωv.

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Fig. 8

Initial (a), penultimate (b), and ultimate (c) radial velocity eigenmode solution of the stability analysis around the mean turbulent flow

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Fig. 9

Radial (a) and axial (b) volume force obtained by minimizing the dominant eigenvalue growth rate of the mean turbulent flow that quenches the part load vortex rope

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Fig. 10

Flow modification distributions due to the ultimate bulk force δC=C¯initial−C¯ultimate (a) and time-averaged velocity profiles of the initial C¯initial and ultimate C¯ultimate flow field in section 1.3 (b) and section 1.75 (c)

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Fig. 11

Monitored pressure coefficients in section S1.3 at the sensor locations 1–4, during the transient control of the part load vortex rope in the elbow geometry

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Fig. 12

Time history of the vortex flow in the turbine elbow draft tube, made visible by the pressure isocontour Cp = −4.1 and the axial velocity distribution on the meridional plane since the bulk force is activated

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Fig. 13

Time-averaged velocity isocontour of C¯R=0, C¯Z=0 and the tangential velocity equal to the velocity of the runner at the end of the tip of the runner cone C¯θ=ωrun·RrunCone for the controlled flow solution, sketching an initial guess appendage as the solution of the flow modification



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