BBAA VI International Colloquium on:Bluff Bodies Aerodynamics & Applications

Milano, Italy, July, 20–24 2008

THE EFFECT OF NONHARMONIC FORCING ON BLUFF-BODYAERODYNAMICS AT A LOW REYNOLDS NUMBER

Efstathios Konstantinidis and Demetri Bouris

Department of Engineering and Management of Energy ResourcesUniveristy of Western Macedonia, Bakola and Sialvera, 50100 Kozani, Greece

e-mails:ekonstantinidis@uowm.gr, dmpouris@uowm.gr

Keywords: Circular cylinder, Forced wake, Nonharmonic forcing, Aerodynamics

1 INTRODUCTION

It is well known that the wake of a bluff-body behaves like a self-excited non-linear fluidoscillator above a critical Reynolds number≈45, giving rise to periodic vortex formation andshedding at the Strouhal frequency [1]. Beyond the criticalpoint further instabilities lead pro-gressively to three-dimensionality and turbulence in the wake as a function of the Reynoldsnumber. When the fluid oscillator is externally forced then the resulting wake flow is depen-dent on the forcing parameters as well as the Reynolds number. For harmonic forcing such asrectilinear cylinder oscillations at any angle to the flow direction, rotational oscillations, inflowfluctuations, etc., the response of the wake becomes a function of the forcing frequency andamplitude because the perturbation imposed can be described by a simple sinusoidal function.The study of harmonically forced bluff-body wakes has received considerable attention overthe last decades with a view for flow and/or vibration control. This requires an understandingof the mechanisms of energy transfer between the fluid and thewake-generating body, whichwould give rise to self-excitation in the corresponding case of flexible or elastically mountedstructures. However, the use of harmonic perturbations poses a limitation on the expected rangeof response characteristics since the harmonic solution isa linear approximation to non-linearsystem dynamics.

The purpose of this paper is to study the effect of nonharmonic forcing on the wake responseof a circular cylinder by means of computational fluid dynamics. In order to force the wake, theinflow velocity is given a periodic nonharmonic fluctuation,

U(t) = Uo

(

1 + α2 sin2(ωt))n

, (1)

whereUo is a reference velocity (m/s),ω is the angular frequency (rad/s), andα is a dimension-less parameter relating to the amplitude of fluctuation. Theindex n appears as an additionalparameter that determines the waveform of the imposed perturbation. Forn = 0, there is noperturbation (steady flow) while forn = 1 the classical harmonic perturbation is attained (si-nusoidal). In order to have comparable results for different nonharmonic perturbations, theα

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Efstathios Konstantinidis and Demetri Bouris

values were selected so that the minimum and maximum inflow velocities were a preset fac-tor of the reference velocityUo. The Reynolds number based on the reference velocity is 180.For the present study the peak-to-peak amplitude was set at 0.65. The frequency of imposedfluctuation was0.84fo wherefo is the natural vortex shedding frequency in the unforced wake.Therefore, the effect of varying the indexn is reported in this paper with all other parametersheld constant. Figure 1 shows the waveforms forn = −1, 1/2, 1/3 and 1 employed for thepresent results. It should be noted that the actual frequency of forcing is twice that noted aboveby virtue of Eq. (1). For the analysis of the results, the wakeresponse is determined from thecomputed time series of the lift and drag forces exerted on the cylinder as well as the patternsof vorticity distribution.

0.00 0.25 0.50 0.75 1.000.0

0.5

1.0

1.5

n = 1 n = 1/2 n = 1 n = 1/3

U(t)

tFigure 1: The perturbation waveforms from Eq. (1) employed in the present study.

2 COMPUTATIONAL SCHEME AND RESULTS

For the computations, the time-dependent volume-averagedNavier–Stokes equations weresolved on a collocated orthogonal curvilinear grid using a SIMPLE-based algorithm with fully-implicit first-order Euler discretisation in time and a bounded second-order upwind schemefor the convection terms [2]. At the lateral boundaries of the mesh symmetry conditions wereapplied, Neumann conditions at the outlet and a no-slip condition at the cylinder wall. Table (1)lists the time-averaged properties of the drag and lift forces in steady flow after the solutionstabilises to a limit cycle for later comparison. The results agree well with other published data,e.g. [3, 4]. The instantaneous solution of the flow field at theend of the nonforced case wasused as an initial condition for the forced flow computations.

Re St CD C ′

LC ′

D

180 0.192 1.334 0.455 0.018

Table 1: Computational results for unforced flow.

Figure (2) shows the Lissajous plots of the drag (CD) and lift (CL) forces in fluctuating flowafter a periodic solution was attained. For all cases, the frequency of the drag force is locked-onwith double the nominal excitation frequency while that of the lift force is locked-on with thenominal excitation frequency. This might be expected basedon the case of harmonic excitationsince the{ω, α} combination falls within the fundamental lock-on range forsinusoidal cylinderoscillations or equivalent inflow fluctuations [5, 6]. This is accompanied by a magnification ofthe fluctuating forces exerted on the cylinder compared to the unforced flow. A notable variation

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Efstathios Konstantinidis and Demetri Bouris

in the Lissajous figures as well as in the peak/trough forces can be observed as a function ofthe indexn (Fig. 2). Forn > 0, a three-knotted figure is formed which is more pronouncedwith decreasingn values. Forn = −1, the two knots disappear and a figure-8 pattern emergesaccompanied by an increase in the drag fluctuation (9%) and a decrease in the lift (15%) whereasthe mean drag increases toCD = 1.470, a 13%-decrease compared to the harmonically forcedflow (n =1).

-1

0

1

0 2 4

-1

0

1

0 2 4

n = 1

CL

n = 1/2

n = 1

CL

CD

n = 1/3

CD

Figure 2: Lissajous plots of the drag and lift force for different perturbation waveforms.

The patterns of vorticity distribution in the near wake for different perturbations at the in-stant corresponding toU(t) ≈ Uo (during the acceleration phase) are shown in Fig. (3). Forharmonic forcing (n =1), the pattern ensuing after shedding of vorticity from thecylinder hasan asymmetrical structure consisting of a single positive vortex and a pair of negative vortices(this results in a mean lift coefficient different from zero). This pattern corresponds to the P+Smode which occurs at two-dimensional conditions [7]. The same pattern occurs for alln > 0but the shed vortices have travelled further downstream with decreasingn values. This mightbe attributed to the quicker vortex shedding induced by the more abrupt changes inU(t) nearthe minimum inflow velocity, which, for harmonically forcedflow, corresponds to the timingof shedding [8]. Forn = −1, however, a complex pattern consisting of two pairs of vorticesensues similar to the known 2P mode [7]. Note that the follow up vortex shed from either sidehas a higher peak vorticity than the initial vortex shed fromthe same side. The flow within thevortex formation region for this case is marked by the clear formation of two vortices in theshear layer separating from the low side of the cylinder.

3 CLOSING REMARKS

The results indicate a substantial effect of the nonharmonic perturbations on the forces ex-erted on the circular cylinder (magnitude and phase) and on the vortex wake dynamics even ata low Reynolds number. More computations are carried out to study the effect in the three-dimensional parameter space{ω, α, n}.

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Efstathios Konstantinidis and Demetri Bouris

x

y

0 2 4 6 8

-2

0

2

n = 1

x

y

0 2 4 6 8

-2

0

2

n = 1/2

x

y

0 2 4 6 8

-2

0

2

n = 1

x

y

0 2 4 6 8

-2

0

2

n = 1/3

Figure 3: Patterns of vorticity distribution in the near wake for different perturbation waveforms. Normalizedvorticity contours:±0.5,±1, . . .±5. Peak positive and negative contours are coded red and bluerespectively.

REFERENCES

[1] P. Huerre and P. A. Monkewitz. Local and global instabilities in spatially developing flows.Annual Review of Fluid Mechanics, 22:473–537, 1990.

[2] D. Bouris, E. Konstantinidis, S. Balabani, D. Castiglia, and G. Bergeles. Design of a novel,intensified heat exchanger for reduced fouling rates.International Journal of Heat andMass Transfer, 48(18):3817–3832, 2005.

[3] R. D. Henderson. Details of the drag curve near the onset of vortex shedding.Physics ofFluids, 7(9):2102–2104, 1995.

[4] C. Norberg. Fluctuating lift on a circular cylinder: review and new measurements.Journalof Fluids and Structures, 17(1):57–96, 2003. 0889-9746.

[5] O. M. Griffin and H. S. Hall. Vortex shedding lock-on and flow control in bluff body wakes- review. Journal of Fluids Engineering, 113:283–291, 1991.

[6] E. Konstantinidis, S. Balabani, and M. Yianneskis. The effect of flow perturbations on thenear wake characteristics of a circular cylinder.Journal of Fluids and Structures, 18:367–386, 2003.

[7] C. H. K. Williamson and A. Roshko. Vortex formation in thewake of an oscillating cylinder.Journal of Fluids and Structures, 2:355–381, 1988.

[8] E. Konstantinidis, S. Balabani, and M. Yianneskis. The timing of vortex shedding in a cylin-der wake imposed by periodic inflow perturbations.Journal of Fluid Mechanics, 534:45–55, 2005.

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