The effect of nonharmonic perturbations on vortex dynamics in bluff-body wakes

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    The effect of nonharmonic perturbations on vortex dynamics in bluff-body wakes

    E. Konstantinidis*and D. Bouris

    Department of Mechanical Engineering, University of Western Macedonia, Kozani 50100, Greece

    SUMMARY

    Numerical simulations of two-dimensional flow about a circular cylinder were carried out in order to

    study the effect of nonharmonic perturbations of the inflow velocity on the vortex dynamics in the wake

    and the fluid forcing on the cylinder for Reynolds numbers in the laminar regime. Two different

    nonharmonic waveforms are examined and the results are compared to the harmonic one. An intriguing

    result is that the dynamic response of the wake (i.e. whether phase-locked or not) can be modified for the

    different waveforms if some reference velocity (e.g. the minimum/maximum velocity or their mean

    value) is employed as the basis for the comparisons. However, if the true time-averaged flow velocity is

    taken into account, then the limits of the lock-on regime fall together on the frequency-amplitude plane.

    The dimensionless coefficient of energy transfer from the fluid to a corresponding oscillating cylinder

    inline with the flow direction is negative in all cases examined but its magnitude depends on theperturbation waveform.

    The scope of the present study is to improve the understanding of the mechanisms of vortex formation in the

    wake of oscillating bluff bodies and the associated energy transfer between the body and the fluid. A practical and

    useful approach to achieve this objective is to consider a fixed body, i.e. a circular cylinder, exposed to time-

    dependent flows since only the relative motion of the body through the fluid matters. Using this approach, it has been

    made possible to explain the lack of self-excitation of vortex-induced inline vibrations when the shedding frequency

    coincides with the natural frequency of an elastically mounted cylinder (Konstantinidis et al., 2005). The advantage

    of this approach is that it allows one to concentrate solely on the vortex dynamics in the wake. An important

    parameter that has not received much attention is the effect of the rate of change of the relative velocity between the

    fluid and the body. Most of the previous work dealt with harmonic waveforms even though an infinite number of

    nonharmonic waveforms are attainable in practice due to amplitude and frequency modulations in the relative motionand/or the fluid forcing.

    A recently published work has illustrated that nonharmonic perturbations of the inflow velocity can generate

    different patterns of phase-locked vortex formation in the wake of a circular cylinder, involving combinations of

    single and/or pairs of vortices, compared to pure harmonic perturbations for the same forcing period and peak-to-

    peak amplitude of the perturbations (Konstantinidis and Bouris, 2009). This previous numerical work has examined

    the effect of varying the perturbation waveforms for a given combination of the perturbation frequency and

    amplitude. An illustrative example of the different patterns of vortex formation is shown in figure 1.

    In this work, we extent the numerical simulations to cover a range of perturbation frequencies and amplitudes

    encompassing the vortex shedding lock-on envelope for two specified waveforms. The numerical solution is based

    on the discretization of the governing equations on an orthogonal curvilinear mesh using the finite-volume method.

    At the inflow boundary of the solution domain, prescribed velocity perturbations are superimposed on a non-zeromean in order to act as an external excitation source. Two complementary waveforms of the time-dependent inflow

    velocity U(t) were generated by

    2

    2

    1 sin ( )( )

    1 cos ( )

    n

    n

    tU t

    t

    + + =

    +

    (1)

    where is the cyclic frequency of the perturbations, is a parameter related to the amplitude of the velocity

    perturbations, sets the mean velocity and the index n determines the waveform. For n = 1 the perturbation

    waveform is a pure harmonic, a case which is employed as the basis for comparisons to the nonharmonic

    perturbations. For the latter case, the index was set at n= 1 and two different waveforms were employed using the

    two formulas in (1).

    *E-mail for correspondence: [email protected]

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    FIGURE 1. Patterns of vortex formation in the cylinder wake induced by different nonhanmonic

    waveforms for the same amplitude and frequency of perturbations at Re0= 180; (a) n= 1, (b) n= 1/5, (c)

    n= -1. See equation (1) for the definition of the nonharmonic waveforms.

    In this work, we examine the effects of varying the frequency and amplitude of the velocity perturbations. This

    was done by varying at constant until the lower and upper limit of the vortex shedding lock-on range was

    reached, then repeating the procedure at different amplitudes. The parameter was adjusted so that the maximum

    inflow velocity was the same in every case considered. This maximum value of the inflow velocity corresponded to

    an instantaneous Reynolds number of 180 in order to limit the simulations to the regime where the wake is expected

    to remain laminar and two-dimensional in the steady flow (Williamson, 1996). It is useful to define a reference

    velocity ( )10 max m2U U U= + in and amplitude ( )1

    max min2U U U = based on the maximum and minimum velocity in

    the waveform. Figure 2 shows the perturbation waveforms employed in the present simulations. The two

    nonharmonic waveforms will be denoted -type and m-type because of their shape. Two important points should be

    noted: (a) the time-averaged mean velocity is not equal to U0 for the nonharmonic waveforms, (b) the velocity

    waveform repeats itself twice in each cycle so that the actual excitation frequency is twice the nominal frequency in

    eq. (1). The energy contained in the harmonic part of the nonharmonic waveforms is 93.5% of the total energy;

    hence, the deviations from a sinusoidal waveform are quite moderate.

    0.0 0.5 1.0 1.5 2.0

    -1

    0

    1

    U(t) U0

    U

    t/T FIGURE2. Waveforms of the forcing drivers employed in the present simulations; solid line: harmonic,

    dashed line: -type nonharmonic, dotted line: m-type nonharmonic.

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    Figure 3 shows a map of the limits of the vortex shedding lock-on regime in the frequencyamplitude plane for

    different perturbation waveforms. Within this regime the vortex formation is phase-locked with the imposed velocity

    perturbations and the shedding frequency is equal to the nominal excitation frequency. In this plot, the excitation

    frequency fe is normalized with the frequency of vortex from a fixed cylinder f0 corresponding to 0 0Re /U D =

    which was computed from the Strouhal-Reynolds number relationship given by Williamson and Brown (2001). The

    perturbation amplitude is normalized with the reference velocity. It can be observed that the different waveformshave a considerable effect on the lock-on map, particularly at the higher amplitudes. The -type perturbations cause

    a shifting of the limits towards lower frequency ratios compared to harmonic perturbations whereas the m-type

    perturbations have the opposite effect. Therefore, it is possible that for a given combination of the perturbation

    frequency and amplitude the wake response can be phase-locked or not depending on the perturbation waveform. For

    example, if 0.56 and 0.4 vortex lock-on occurs for the -type waveform but the wake is not

    phase-locked for the other waveforms. The effect of different perturbation waveforms becomes more pronounced

    with increasing perturbation amplitude.

    0/

    ef f =

    0/U U =

    The above intriguing result can be resolved if we modify the independent variables to take into account the fact

    that the true time-averaged flow velocity in not equal to U0 and, hence, both the perturbation frequency and

    amplitude ratios need be corrected. Let be the true time-averaged flow velocity and*0U

    *

    0f the expected shedding

    frequency at the true mean velocity. In the modified lock-on map, shown in figure 4, the synchronization envelopesfor different nonharmonic waveforms collapse on top of that for harmonic perturbations. However, it should be

    pointed out that the Reynolds number based on the true mean flow velocity is not the same at constant amplitude for

    different waveforms. Hence, there is competition between two different effects, one due to the perturbation

    waveform and the other due to the Reynolds number. It is impossible to separate these two effects.

    The patterns of vortex formation were also modified by different perturbation waveforms. As a consequence,

    other characteristics of the cylinder wake, such as the mean drag force coefficient, exhibited considerable

    modifications for different waveforms (results not shown here for economy of space). The dimensionless coefficient

    of energy transfer CEbetween the fluid and the cylinder for the corresponding problem where the incident flow is

    steady and the cylinder is oscillating inline with the flow,

    ( ) (*0 ( ) ( )d E DT

    C U U t C t t =

    )T (2)

    where CD(t) is the time-dependent drag force coefficient and the integration is carried out over a cycle of oscillation

    (the period of the flow perturbations here) was also computed from the data. The results indicate that the energy

    transfer is always (i.e. for all combinations of the frequency and amplitude of the perturbations and all different

    waveforms examined) negative which implies the lack of excitation of vortex-induced inline vibrations of elastically-

    mounted cylinders at low Reynolds numbers. However, the CE magnitude is quite sensitive to the type of the

    perturbation waveform. Some work in progress, indicates that this dependence cannot be attributed to the implied

    difference in the mean flow velocity (Reynolds number effect) solely.

    As a conclusion, different nonharmonic perturbation waveforms have some effect on the vortex dynamics in

    the wake and the forcing on the cylinder which becomes more pronounced with increasing perturbation amplitude.

    For the perturbation waveforms employed in the present study, these effects can be attributed, partially at least, to the

    implied change in the true mean velocity, i.e. to the effect of Reynolds number.

    REFERENCES

    Konstantinidis, E., Balabani, S. and Yianneskis, M. (2005) The timing of vortex shedding in a cylinder wake

    imposed by periodic inflow perturbations. Journal of Fluid Mechanics 543: 44-55.

    Konstantinids, E., Balabani, S. and Yianneskis, M. (2007) Bimodal vortex shedding in a perturbed cylinder wake.

    Physics of Fluids 19: 011701, 1-4.

    Konstantinidis, E. and Bouris, D. (2009) Effect of nonharmonic forcing on bluff-body vortex dynamics. Physical

    Review E 79(4): 045303.

    Williamson, C. H. K. (1996). Vortex dynamics in the cylinder wake. Annual Review of Fluid Mechanics 28: 477-

    539.

    Williamson, C. H. K. and Brown, G. L. (1998) A series in 1 Re to represent the Strouhal-Reynolds numberrelationship of the cylinder wake. Journal of Fluids and Structures 12: 1073-1085.

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    0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

    0.0

    0.1

    0.2

    0.3

    0.4

    0.5

    U

    U0

    fe/

    f0

    FIGURE3. Map of the lock-on limits for different perturbation waveforms. Grey shading corresponds to

    the harmonic waveform, \\\\ -type , //// m-type.

    0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

    0.0

    0.1

    0.2

    0.3

    0.4

    U

    U*

    0

    fe/f *

    0

    FIGURE 4. Modified map of the lock-on limits for different perturbation waveforms. Grey shading

    corresponds to the harmonic waveform, \\\\ -type , //// m-type.