14th Australasian Fluid Mechanics Conference Adelaide University, Adelaide, Australia 10-14 December 2001

Frequency and Amplitude Effects in the

J. Young and J.C.S

School of Aerospace and MechaUniversity College, University of

Australian Defence ForceCanberra, ACT, 2600, AU

Abstract The flow over a 2D airfoil, oscillated sinusoidally in plunge, is simulated using a compressible Navier Stokes solver at Reynolds number 30,000. Close agreement is obtained between numerically simulated wake structures and experimental wake visualisations in the literature. Thrust of the airfoil and detailed wake structures are shown to be strongly dependent on both the Strouhal number and the reduced frequency of the plunge oscillation, at this Reynolds number. The findings are postulated to be the result of a vortex shedding frequency lock-in phenomenon, similar to that observed for transversely vibrating circular cylinders. Nomenclature a Single sided plunge amplitude c Airfoil chord f Plunge frequency in Hz h Non-dimensional plunge amplitude a/c k Plunge reduced frequency ω c/2U∞ St Strouhal number 2af /U∞ = 2kh/π U∞ Free stream velocity ω Angular frequency 2π f Introduction Oscillating airfoils have been extensively studied in a variety of contexts. Experimental and numerical studies have explored the dynamics of the wake and vortex roll-up (Katz and Weihs [1]), the dynamic stall phenomenon (eg. Choudhuri & Knight [2], Ekaterinaris & Platzer [3], Isogai et al [4]), details of fish and marine mammal propulsion (eg. Lighthill [5], Triantafyllou et al [6]), the use of oscillating airfoils in flow control and flow reattachment (Lai et al [7], Tuncer et al [8]), and wing and turbomachinery blade flutter. The use of flapping wings for generation of lift and propulsion has also gained gained recent attention with application to Micro Air Vehicles (eg. Shyy et al [9]). Garrick [10] determined the propulsive efficiency of a plunging and pitching flat-plate airfoil as a function of the oscillation frequency, with the assumption of potential flow and small amplitude oscillation. Pure plunging airfoil motion has received relatively less attention than pitching motion. Freymuth [11], Jones et al [12] and Lai & Platzer [13] show that the wakes of plunging airfoils can be characterized as drag-producing, neutral, or thrust-producing depending on the plunge frequency and amplitude. Several analyses have postulated that the Strouhal number of the oscillation is the key parameter determining the thrust coefficient (Ramamurti & Sandberg [14], Jones et al [12]), and therefore the wake structures. However, details of wake structure formation, and lift and thrust generation, are not well understood. The objective of this paper is to explore the effects of plunging frequency and amplitude on the wake structures of an airfoil undergoing pure plunging motion, y = a cos(ω t).

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Wake of a Plunging Airfoil

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erical Method unsteady, viscous, compressible 2D Navier Stokes equations solved in generalised coordinates on a 537×81 element tured C-grid (187 points on the airfoil), which is deformed nd the airfoil to introduce the plunging motion. The thin- approximation [15] is implemented, as is the Baldwin-ax turbulence model [15] for (Reynolds averaged) fully lent flow calculations. The flow solver uses an implicit -order accurate Osher upwind-biased flux difference ting scheme to solve the strong conservation law form of the tions. Details of the method may be found in Tuncer et al . res 1 to 4 show experimental wake visualisations reproduced Lai & Platzer [13] (upper frames), and the corresponding laminar (middle frames) and fully turbulent (lower frames) erical visualisations generated with a particle tracing method. experimental results were generated in a water tunnel at Re = 00, and the minimal variation expected in results due to the rence in simulated and experimental conditions is discussed oung and Lai [17]. In any case only qualitative agreement of ake structures is sought here.

e 1. Experimental k = 3.93, h = 0.0125 (upper), Numerical k = 4.0, .0125, laminar (middle) and turbulent (lower). ach case the numerical method approximates the wake viour closely for the fully laminar calculations, but poorly < about 0.15) to moderately well (kh > 0.15) for the fully lent calculations. The airfoil in Figure 1 is producing drag,

as is apparent from the upstream-tilting of the vortex pairs, and from averaged-flow numerical calculations. We also observe two distinct vortices shed per half-cycle of airfoil motion. Figure 2 shows a neutral-thrust condition, with un-tilted vortex pairs and again two vortices per half-cycle. Figure 3 shows a thrust-producing case, with downstream.-tilting vortex pairs and only one large vortex per half-cycle. Figure 4 shows the flow at twice the frequency and about half the amplitude of figure 2, where now the airfoil is drag-producing, and the wake structure is a single shed vortex per half-cycle.

Figure 2. Experimental k = 3.93, h = 0.025 (upper), Numerical k = 4.0, h = 0.02, laminar (middle) and turbulent (lower).

Figure 3. Experimental k = 3.93, h = 0.05 (upper), Numerical k = 4.0, h = 0.05, laminar (middle) and turbulent (lower).

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e 4. Experimental k = 7.85, h = 0.0125 (upper), Numerical k = 8.0, .0125, laminar (middle) and turbulent (lower).

ay be seen from figures 1 to 4 above that the numerical od qualitatively captures the behaviour of the wake quite ly when the flow is assumed to be fully laminar. The ining results in this paper build on this good agreement to

ain the observed wake structures in terms of the small-scale ls of the flow at the trailing edge.

e 5. Numerical laminar k = 8.0, h = 0.00625 (upper), k = 4.0, h = 5 (middle), k = 2.0, h = 0.025 (lower).

e Structure Dependence on Plunging Frequency Amplitude parison of the experimental (upper) frames of figures 2 and ows that for the same kh (ie. the same St), the wake can bit significantly different structures. The fully laminar

calculations have also qualitatively captured this difference in wake behaviour. The explicit dependence of the wake on both k and h is more apparent in figure 5, which shows three (laminar) wakes at the same kh. The three frames are plotted at the same equivalent scale, normalised against a. As k is decreased from 8 to 4 to 2, we se a change from a single, to two, and then multiple vortices shed per half-cycle of airfoil motion. Figure 6 plots numerically obtained thrust coefficient results, as well as the thrust predicted by Garrick [10] for a flat plate undergoing pure plunge in potential flow, which in present notation is CTmean = 4π (kh)2 (F2(k)+G2(k)), where F and G are the real and imaginary components of the Theodorsen Function. For comparison, the Garrick results are corrected for the numerically predicted drag of a stationary airfoil. Figure 6 shows a quantitative dependence of the thrust coefficient on both k and h, to a degree not predicted by the Garrick analysis.

Figure 6. Thrust coefficient variation with kh, numerical laminar results and Garrick linearised analysis. Natural Shedding Frequency Phase-Locking The first clue to the origin of this explicit dependence of the wake structure and the thrust coefficient on k and h can be gained by noting that at the Reynolds number under consideration, the airfoil exhibits a natural shedding frequency, in the manner of bluff bodies. This is shown both experimentally and numerically for a stationary airfoil in figure 7.

Figure 7. Natural vortex shedding for a stationary airfoil, experimental (upper), numerical laminar (lower).

For the stationary airfoil there is a separation bubble of temporally-varying extent on either side of the trailing edge. Flow separation upstream of the trailing edge is also apparent when the airfoil is plunging. Effectively the trailing edge is no longer sharp, but rather there is a ‘blunt’ region of reversed flow

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nd which the remaining flow travels. This effect can be ly seen in figure 8. Consequently we see natural shedding, lar to that for bluff bodies, and for which the shedding ency is dependent on the extent of the separation region. presence of two frequencies in the flow (natural shedding ency and plunge frequency) would be expected to give rise ode competition and phase-locking, similar to that seen in akes of circular cylinders [18].

e 8. Natural vortex shedding around a persistent trailing edge x, numerical laminar results, k = 20.0, h = 0.001. Trailing edge ion vs time is indicated in right hand graph, two complete plunge s shown.

the flow here does indeed undergo phase-locking may be in figures 9, 10 and 11, where in each case the upper frame s the wake dominated by vortex shedding at the plunging ency, and the lower frame shows the wake dominated by

ral shedding when the plunge amplitude is reduced.

Figure 9. Numerical laminar, k = 20.0, h = 0.005 (upper), k = 20.0, h = 0.001 (lower).

Figure 10. Numerical laminar, k = 8.0, h = 0.0125 (upper), k = 8.0, h = 0.001 (lower).

Figure 11. Numerical laminar, k = 4.0, h = 0.025 (upper), k = 4.0, h = 0.001 (lower).

Circular cylinder results [18] indicate that whether or not the flow becomes phase-locked for a given k depends on the power input (determined by kh) at that frequency, with higher power inputs resulting in phase-locking. This is also observed in figures 9 to 11 above. The wake structures shown in figures 1 to 5 can now be explained in terms of the phase-locking phenomenon. At a kh high enough to induce phase-locking, as k is increased from zero, the natural shedding frequency becomes locked, not to the plunge frequency, but to a higher harmonic, so that an integral number of vortices are shed per half-cycle of airfoil motion. As k is increased further, the natural shedding becomes locked to the

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ge frequency itself, and we see only a single vortex shed per cycle.

clusions ils of the wake structure of a plunging airfoil have been ly replicated using a Navier-Stokes numerical method, for a

e of plunging frequency k and amplitude h. Both rimental and numerical results indicate that the wake ture is strongly dependent on both k and h, for the Reynolds ber range under study. This dependence is postulated to be esult of phase-locking between the plunge frequency, and irfoil natural shedding frequency at that Reynolds number.

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