Mode-locking and quasi-periodicity in the bifurcation behaviour of a vortex shedding model

N. W. Mureithi, R. Masaki, Shigehiko Kaneko, T. Nakamura

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

Vortex-shedding at low Reynolds numbers in the wake of an oscillating cylinder has been shown to exhibit characteristics universal to nonlinear dynamical systems with competing frequencies, generically referred to as quasi-periodic systems, The (1:1) lock-on phenomenon is the best known and, to date, of most interest in engineering applications. At lock-in, frequency entrainment occurs so that the characteristic frequencies of the two coupled oscillators coincide. The 1:1 lock-in is one in a potentially large (infinite) number of possible m:n (frequency ratio) mode-lockings for nonlinear systems with competing frequencies. In this paper, the lock-in behavior of an analytical model for vortex-shedding at high Reynolds number is studied, An analysis of the model shows that the frequency characteristics of a cylinder under forced excitation are dominated by m:n (rational frequency ratio) mode-locking as the flow velocity (shedding frequency) is varied. Hence, when the cylinder is free to oscillate, vortex shedding is no longer expressed by a constant Strouhal number. Instead the shedding frequency increases in discontinuous steps. Each step marks a jump from one mode-locked m:n state to the next. Although the model corresponds to tube excitation in the lift direction, it is anticipated that the observed behavior is also reflected in the drag direction in view of the fact that vortex shedding interacts with the cylinder in both directions simultaneously. The results may explain the observation by King (1977) that "... the ratio of cylinder frequency to wake frequency apparently 'slips' into convenient numbers ....7:2, 3:1, 7:3, 13:6 ......." in the drag direction, The results are practically significant in view of a recent temperature probe failure in a Japanese fast breeder reactor caused by drag-direction vortex excitation. The failure was attributed to a mode locked resonance (symmetric vortex shedding) which commences at a 4:1 frequency ratio.

Original languageEnglish
Pages (from-to)19-24
Number of pages6
JournalAmerican Society of Mechanical Engineers, Pressure Vessels and Piping Division (Publication) PVP
Volume363
Publication statusPublished - 1998 Dec 1
Externally publishedYes

Fingerprint

Laser mode locking
Vortex shedding
Drag
Reynolds number
Oscillating cylinders
Nonlinear dynamical systems
Strouhal number
Breeder reactors
Time varying systems
Flow velocity
Nonlinear systems
Analytical models
Vortex flow

ASJC Scopus subject areas

  • Industrial and Manufacturing Engineering
  • Mechanical Engineering

Cite this

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title = "Mode-locking and quasi-periodicity in the bifurcation behaviour of a vortex shedding model",
abstract = "Vortex-shedding at low Reynolds numbers in the wake of an oscillating cylinder has been shown to exhibit characteristics universal to nonlinear dynamical systems with competing frequencies, generically referred to as quasi-periodic systems, The (1:1) lock-on phenomenon is the best known and, to date, of most interest in engineering applications. At lock-in, frequency entrainment occurs so that the characteristic frequencies of the two coupled oscillators coincide. The 1:1 lock-in is one in a potentially large (infinite) number of possible m:n (frequency ratio) mode-lockings for nonlinear systems with competing frequencies. In this paper, the lock-in behavior of an analytical model for vortex-shedding at high Reynolds number is studied, An analysis of the model shows that the frequency characteristics of a cylinder under forced excitation are dominated by m:n (rational frequency ratio) mode-locking as the flow velocity (shedding frequency) is varied. Hence, when the cylinder is free to oscillate, vortex shedding is no longer expressed by a constant Strouhal number. Instead the shedding frequency increases in discontinuous steps. Each step marks a jump from one mode-locked m:n state to the next. Although the model corresponds to tube excitation in the lift direction, it is anticipated that the observed behavior is also reflected in the drag direction in view of the fact that vortex shedding interacts with the cylinder in both directions simultaneously. The results may explain the observation by King (1977) that {"}... the ratio of cylinder frequency to wake frequency apparently 'slips' into convenient numbers ....7:2, 3:1, 7:3, 13:6 .......{"} in the drag direction, The results are practically significant in view of a recent temperature probe failure in a Japanese fast breeder reactor caused by drag-direction vortex excitation. The failure was attributed to a mode locked resonance (symmetric vortex shedding) which commences at a 4:1 frequency ratio.",
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AU - Mureithi, N. W.

AU - Masaki, R.

AU - Kaneko, Shigehiko

AU - Nakamura, T.

PY - 1998/12/1

Y1 - 1998/12/1

N2 - Vortex-shedding at low Reynolds numbers in the wake of an oscillating cylinder has been shown to exhibit characteristics universal to nonlinear dynamical systems with competing frequencies, generically referred to as quasi-periodic systems, The (1:1) lock-on phenomenon is the best known and, to date, of most interest in engineering applications. At lock-in, frequency entrainment occurs so that the characteristic frequencies of the two coupled oscillators coincide. The 1:1 lock-in is one in a potentially large (infinite) number of possible m:n (frequency ratio) mode-lockings for nonlinear systems with competing frequencies. In this paper, the lock-in behavior of an analytical model for vortex-shedding at high Reynolds number is studied, An analysis of the model shows that the frequency characteristics of a cylinder under forced excitation are dominated by m:n (rational frequency ratio) mode-locking as the flow velocity (shedding frequency) is varied. Hence, when the cylinder is free to oscillate, vortex shedding is no longer expressed by a constant Strouhal number. Instead the shedding frequency increases in discontinuous steps. Each step marks a jump from one mode-locked m:n state to the next. Although the model corresponds to tube excitation in the lift direction, it is anticipated that the observed behavior is also reflected in the drag direction in view of the fact that vortex shedding interacts with the cylinder in both directions simultaneously. The results may explain the observation by King (1977) that "... the ratio of cylinder frequency to wake frequency apparently 'slips' into convenient numbers ....7:2, 3:1, 7:3, 13:6 ......." in the drag direction, The results are practically significant in view of a recent temperature probe failure in a Japanese fast breeder reactor caused by drag-direction vortex excitation. The failure was attributed to a mode locked resonance (symmetric vortex shedding) which commences at a 4:1 frequency ratio.

AB - Vortex-shedding at low Reynolds numbers in the wake of an oscillating cylinder has been shown to exhibit characteristics universal to nonlinear dynamical systems with competing frequencies, generically referred to as quasi-periodic systems, The (1:1) lock-on phenomenon is the best known and, to date, of most interest in engineering applications. At lock-in, frequency entrainment occurs so that the characteristic frequencies of the two coupled oscillators coincide. The 1:1 lock-in is one in a potentially large (infinite) number of possible m:n (frequency ratio) mode-lockings for nonlinear systems with competing frequencies. In this paper, the lock-in behavior of an analytical model for vortex-shedding at high Reynolds number is studied, An analysis of the model shows that the frequency characteristics of a cylinder under forced excitation are dominated by m:n (rational frequency ratio) mode-locking as the flow velocity (shedding frequency) is varied. Hence, when the cylinder is free to oscillate, vortex shedding is no longer expressed by a constant Strouhal number. Instead the shedding frequency increases in discontinuous steps. Each step marks a jump from one mode-locked m:n state to the next. Although the model corresponds to tube excitation in the lift direction, it is anticipated that the observed behavior is also reflected in the drag direction in view of the fact that vortex shedding interacts with the cylinder in both directions simultaneously. The results may explain the observation by King (1977) that "... the ratio of cylinder frequency to wake frequency apparently 'slips' into convenient numbers ....7:2, 3:1, 7:3, 13:6 ......." in the drag direction, The results are practically significant in view of a recent temperature probe failure in a Japanese fast breeder reactor caused by drag-direction vortex excitation. The failure was attributed to a mode locked resonance (symmetric vortex shedding) which commences at a 4:1 frequency ratio.

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