Does flutter prevent drag reduction by reconfiguration?

Abstract

The static reconfiguration of flexible beams exposed to transverse flows is classically known to reduce the drag these structures have to withstand. But the more a structure bends, the more parallel to the flow it becomes, and flexible beams in axial flows are prone to a flutter instability that is responsible for large inertial forces that drastically increase their drag. It is, therefore, unclear whether flexibility would still alleviate, or on the contrary enhance, the drag when flapping occurs on a reconfiguring structure. In this article, we perform numerical simulations based on reduced-order models to demonstrate that the additional drag induced by the flapping motion is almost never significant enough to offset the drag reduction due to reconfiguration. Isolated and brief snapping events may transiently raise the drag above that of a rigid structure in the particular case of heavy, moderately slender beams. But apart from these short peak events, the drag force remains otherwise always significantly reduced in comparison with a rigid structure.

Keywords: drag reduction; fluid–structure interactions; flutter; instability; reconfiguration.

Figure 1.

(a) Side view of the deforming structure. (b) Examples of geometries of two undeformed structures with different cross-section shapes.

Figure 2.

Critical velocity u_c as a function of the mass ratio β, for λ=10 (solid line), λ=10² (dashed line) and λ=10³ (dotted line).

Figure 3.

Critical Cauchy number C_Y,c as a function of the mass ratio β and static equilibrium shape at the stability threshold for two specific values of the mass ratio, for λ=10 (solid line), λ=10² (dashed line) and λ=10³ (dotted line).

Figure 4.

Vertical amplitude of flapping at the tip (blue) and standard deviation (orange) versus the reduced velocity u, for β=0.5, λ=10. Examples of deformation modes are shown in the thumbnails for u=16.4 (square, static regime (S)), u=22.7 (circle, periodic regime (P)) and u=67.3 (triangle, non-periodic regime (NP)). Static equilibrium shape superimposed (dashed lines).

Figure 5.

Vertical amplitude of flapping at the tip (blue) and standard deviation (orange) versus the reduced velocity u, for β=0.5, λ=10³. Examples of deformation modes are shown in the thumbnails for u=20.0 (square, static regime (S)), u=59.0 (circle, periodic regime (P)) and u=78.0 (triangle, periodic regime also). Static equilibrium shape superimposed (dashed lines).

Figure 6.

Vertical amplitude of flapping at the tip (blue) and standard deviation (orange) versus the reduced velocity u, for β=0.1, λ=10. Examples of deformation modes are shown in the thumbnails for u=5.2 (square, static regime (S)), u=13.2 (circle, periodic regime (P)) and u=36.7 (triangle, non-periodic regime (NP)). Static equilibrium shape superimposed (dashed lines).

Figure 7.

Reconfiguration number $\mathcal{R}$ versus the Cauchy number C_Y, time average (orange) and maximum (blue), for β=0.5, λ=10. Static reconfiguration number (black). The same examples of deformation modes as in figure 4 are shown in the thumbnails, corresponding, respectively, to C_Y=2.69×10³ (square, static regime (S)), C_Y=5.15×10³ (circle, periodic regime (P)) and C_Y=4.53×10⁴ (triangle, non-periodic regime (NP)). Static equilibrium shape superimposed (dashed line). (b) Simply a zoom of the lower right corner of (a).

Figure 8.

Reconfiguration number $\mathcal{R}$ versus the Cauchy number C_Y, time average (orange) and maximum (blue), for β=0.5, λ=10³. Static reconfiguration number (black). The same examples of deformation modes as in figure 5 are shown in the thumbnails, corresponding, respectively, to C_Y=4.00×10⁵ (square, static regime (S)), C_Y=3.48×10⁶ (circle, periodic regime (P)) and C_Y=6.08×10⁶ (triangle, periodic regime also). Static equilibrium shape superimposed (dashed line). (b) Simply a zoom of the lower right corner of (a).

Figure 9.

Reconfiguration number $\mathcal{R}$ versus the Cauchy number C_Y, time average (orange) and maximum (blue), for β=0.1, λ=10. Static reconfiguration number (black). The same examples of deformation modes as in figure 6 are shown in the thumbnails, corresponding, respectively, to C_Y=2.70×10² (square, static regime (S)), C_Y=1.74×10³ (circle, periodic regime (P)) and C_Y=1.35×10⁴ (triangle, non-periodic regime (NP)). Static equilibrium shape superimposed (dashed line).

Figure 10.

Time series of the reconfiguration number in the non-periodic regime of case β=0.1, λ=10, C_Y=1.35×10⁴ (equivalently u=36.7) corresponding to the thumbnail shown in figures 6 and 9. Level of the static reconfiguration number drawn for comparison (dashed line). The time interval displayed corresponds to the whole simulation, apart from the transient regime. Largest snapping event at t_snap=1.019 (triangle). The shape of the structure at t_snap is shown in the thumbnail (solid line), along with the static shape (dashed line) and the average shape (dotted line).

Figure 11.

Linear stability thresholds obtained with the full equation (3.6) for λ=10³ (dotted line), with the equation relative to the axial configuration (3.7) (dashed dotted line) and with the asymptotic equation (B 1) for λ=10³ (solid line).