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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
S
UMMARY
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 the perturbation 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 motion and/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-zero mean in order to act as an external excitation source. Two complementary waveforms of the time-dependent inflow velocity
U
(
t
) were generated by
22
1 sin ( )( )1 cos ( )
nn
t U t t
β α ω β α ω
⎧ ⎡ ⎤+ +⎪ ⎣ ⎦= ⎨⎡ ⎤⎪ − +⎣ ⎦⎩
(1) where
ω
is the cyclic frequency of the perturbations,
α
is a parameter related to the amplitude of the velocity perturbations,
β
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: ekonstantinidis@uowm.gr
F
IGURE
1. Patterns of vortex formation in the cylinder wake induced by different nonhanmonic waveforms for the same amplitude and frequency of perturbations at Re
0
= 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
ω
at constant
α
until the lower and upper limit of the vortex shedding lock-on range was reached, then repeating the procedure at different amplitudes. The parameter
β
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 m2
U U U
= +
in
and amplitude
( )
1max min2
U 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
ω
-type and m-type because of their shape. Two important points should be noted: (a) the time-averaged mean velocity is not equal to
U
0
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.
0.00.51.01.52.0-101
U
(
t
) –
U
0
∆
U t
/
T
F
IGURE
2. Waveforms of the forcing drivers employed in the present simulations; solid line: harmonic, dashed line:
ω
-type nonharmonic, dotted line: m-type nonharmonic.
Figure 3 shows a map of the limits of the vortex shedding lock-on regime in the frequency–amplitude 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
f
e
is normalized with the frequency of vortex from a fixed cylinder
f
0
corresponding to
0 0
Re /
U D
ν
=
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 waveforms have a considerable effect on the lock-on map, particularly at the higher amplitudes. The
ω
-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
ω
-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
/
e
f 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
U
0
and, hence, both the perturbation frequency and amplitude ratios need be corrected. Let be the true time-averaged flow velocity and
*0
U
*0
f
the expected shedding frequency at the true mean velocity. In the modified lock-on map, shown in figure 4, the synchronization envelopes for 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
C
E
between 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
C
D
(
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
C
E
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.
R
EFERENCES
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 number relationship of the cylinder wake. Journal of Fluids and Structures 12: 1073-1085.
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.40.00.10.20.30.40.5
∆
U U
0
f
e
/
f
0
F
IGURE
3. Map of the lock-on limits for different perturbation waveforms. Grey shading corresponds to the harmonic waveform, \\\\
ω
-type , //// m-type.
0.40.50.60.70.80.91.01.11.21.31.40.00.10.20.30.4
∆
U
U
*0
f
e
/
f
*0
F
IGURE
4. Modified map of the lock-on limits for different perturbation waveforms. Grey shading corresponds to the harmonic waveform, \\\\
ω
-type , //// m-type.

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