Dependence on aspect ratio of symmetry breaking for oscillating foils: implications for flapping flight

Abstract

Using two-dimensional direct numerical simulations, we investigate the flow in a fluid of kinematic viscosity ${\it\nu}$

and density

${\it\rho}$

around elliptical foils of density

${\it\rho}_{s}$

with major axis

$c$

and minor axis

$b$

for three different aspect ratios:

$AR=b/c=1$

(a circle);

$AR=0.5$

; and

$AR=0.1$

. The vertical location of these foils

$y_{s}(t)=A\sin (2{\rm\pi}f_{0}t)$

oscillates with amplitude

$A$

and frequency

$f_{0}$

in two distinct ways: ‘pure’ oscillation, where the foils are constrained to remain in place; and ‘flying’ oscillation, where horizontal motion is allowed. We simulate the flow for a range of the two appropriate control parameters, the non-dimensional amplitude, or Keulegan–Carpenter number

$KC=2{\rm\pi}A/c$

, and the non-dimensional frequency, or Stokes number

${\it\beta}=f_{0}c^{2}/{\it\nu}$

. We observe three distinct patterns of asymmetry, labelled ‘S-type’ for synchronous asymmetry, ‘

$\text{QP}_{\text{H}}$

-type’ and ‘

$\text{QP}_{\text{L}}$

-type’ for quasi-periodic asymmetry at sufficiently high and sufficiently low (i.e.

$AR=0.1$

) aspect ratios, respectively. These patterns are separated at the critical locus in

$KC$

–

${\it\beta}$

space by a ‘freezing point’ where the two incommensurate frequencies of the QP-type flows combine, and we show that this freezing point tends to occur at smaller values of

$KC$

as

$AR$

decreases. We find for the smallest aspect ratio case (

$AR=0.1$

) that the transition to asymmetry, for all values of

$KC$

, occurs for a critical value of an ‘amplitude’ Stokes number

${\it\beta}_{A}={\it\beta}(KC)^{2}=4{\rm\pi}^{2}f_{0}A^{2}/{\it\nu}\simeq 3$

. The

$\text{QP}_{\text{L}}$

-type asymmetry for

$AR=0.1$

is qualitatively different in physical and mathematical structure from the

$\text{QP}_{\text{H}}$

-type asymmetry at higher aspect ratio. The flows at the two ends of the ellipse become essentially decoupled from each other for the

$\text{QP}_{\text{L}}$

-type asymmetry, the two frequencies in the horizontal force signature being close to the primary frequency, rather than twice the primary frequency as in the

$\text{QP}_{\text{H}}$

-type asymmetry. Furthermore, the associated coefficients arising from a Floquet stability analysis close to the critical thresholds are profoundly different for low aspect ratio foils. Freedom to move slightly suppresses the transition to S-type asymmetry, and for certain parameters, if a purely oscillating foil subject to S-type asymmetry is released to move, flow symmetry is rapidly recovered due to the negative feedback of small horizontal foil motion. Conversely, for the ‘higher’ aspect ratios, the transition to

$\text{QP}_{\text{H}}$

-type asymmetry is encouraged when the foil is allowed to move, with strong positive feedback occurring between the shed vortices from successive oscillation cycles. For

$AR=0.1$

, freedom to move significantly encourages the onset of asymmetry, but the newly observed ‘primary’

$\text{QP}_{\text{L}}$

-type asymmetry found for pure oscillation no longer occurs when the foil flies, with S-type asymmetry leading ultimately to locomotion at a constant speed occurring all along the transition boundary for all values of

$KC$

and

${\it\beta}$

.