Twenty-First Symposium on Naval Hydrodynamics (1997) / Chapter Skim
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Low-Dimensional Modeling of Flow-Induced Vibrations via Proper Orthogonal Decomposition
Low-Dimensional Modeling of Flow-Induced Vibrations via Proper Orthogonal Decomposition
Pages 605-621
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From page 605... ...
We first simulate the 2-d and 3-d wakes behind vibrating cylinders arid cables, arid then use the method of snapshots to compute the most energetic eigenmodes of these wakes. We examine the eigenmode energy decay versus mode number, and discuss the possibility of constructing low-dimensional dynamic models to simulate and predict the behavior of these systems.
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From page 606... ...
The cable has mass per unit length m and tension T To maintain a mean displacement, the cable is lightly elastically supported by linear springs with spring constant k, giving a natural frequency of a" = I
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From page 607... ...
For each simulation, the cable initial position Ad velocity are set and the simulations are run for at least 20 shedding cycles, or until the statistics are relatively stationary 2.3 POD Analysis To get a more general indication of the structure of a particular wake, eve use proper orthogonal deco,=position. Proper orthogonal decomposition (POD)
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From page 608... ...
Koopmann [93 conducted a series of experiments at sc~reral Reynolds n',mbere measuring the flow behind cylinders oscillating in the crossbow direction. He considered cro~flow vibration amplitude from '7/d = 0.05 to q/d = 1, and he varied the vibration frequency above and below the Vortex shedding frequency of the fixed cylinder at that Reynolds number.
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From page 609... ...
a~ pears stationary and straight. Figures 5, 6, 7 and 8 show the six most energetic eigenmodes for the fixed cylinder and forced vibration cases wf/~0 = 0.G, w'/wO = 0.8, and w'/Loo = 1.4 respectively.
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From page 610... ...
. The classification of the eigenmode pairs follow that of the fixed cylinder, however in this forced case the energy of each eigenmode is subst.an610
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From page 611... ...
ection of eigenmodes can be selected by examining the eigenmode energy spectrum. This is plotted for our fixed cylinder and forced vibration cases in Figure 12.
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From page 612... ...
The standing wave and traveling wave flow-induced vibration responses were generated by placing the initial cable position ~ the form of a standing wave and traveling wave, and then running the simulation for several shedding cycles, at which point a time-periodic state was reached. The cable tension was selected so that vibrations of wavelength l/d = 12.6 would respond to the forcing frequency (estimated from the fixed cylinder Strouhal number, i.e.
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From page 613... ...
Figures 13 and 14 show a top view and perspective view of equal and oppm site levels of spanwise vorticity (~ = :~0.2) for the standing wave cable wane and traveling wave cable wake respectively (for L/d = 12.6 wavelength vibrate lion case)
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From page 614... ...
The top view Id perspective view of equal "d opposite levels of spanwise vorticity (w, = +0.2) for the the Re=200 ~owinduced vibration wake is shown in Figure 16.
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From page 615... ...
Because of the slower eigenmode energy decay rate at Re=200, we plot eigenmodes 1, 4, 8 and 16. Starting with the standing wave wake at Re=lOl)
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From page 616... ...
Re-100 haves wave ___ Re-200 0 20 40 GO 00 1m Mode Figure 18: Eigenmode energy fraction versus mode number for Ad Ilow-induced vibrations, Re=100 arid Re=200. 5 Discussion In this paper we conducted direct numerical simulations of 2-d and Ad flows over oscillating cylinders and flexible cables, Ad then performed proper orthogonal decomposition analyses on the waltes.
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From page 617... ...
~ i - ~ ~ al . O a' 0 ~ 0 ~ 0 ~ :~: c it, ma '—B— - ~ : Figure 19: Eigenmodc 1 for Re=100 standing wave.
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From page 618... ...
~ ~1 0 a' 0 o' 0 ~ 0 u - ~ -- ~ - 1 - - ~ ~ o ~ o ~ ~ Figure 23: Eigenmode 1 for Re=100 traveling wave. mode 3 < O ~ O ~ O ~ O .
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From page 619... ...
O ~ AS _ _ O ~ .~_ Figure 29: Eigenmode 8 for Re=200 flow induced vibration wake. 0\ mode-16 O ~ O it' ~3 ~16 O 619 Figure 30: Eigenmode 16 for Re=200 flow induced iteration wale.
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From page 620... ...
Gharib. An experimental study of the parallel and oblique vortex shedding from circular cylinders.
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From page 621... ...
The oscillation tends to control and possibly strengthen non-oblique vortex shedding, when locked on, but the threedimensional effect of the phase variation limits this to finite length spanwise cells of alternating sign which might be expected to weaken the vortex shedding. Can any observations be made from your three-dimensional results about the lock-in boundaries (oscillation amplitude vs.
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