jats:pUsing two-dimensional direct numerical simulations, we investigate the flow in a fluid of kinematic viscosity jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline1" />jats:tex-math$\nu$</jats:tex-math></jats:alternatives></jats:inline-formula> and density jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline2" />jats:tex-math$\rho$</jats:tex-math></jats:alternatives></jats:inline-formula> around elliptical foils of density jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline3" />jats:tex-math$\rho s$</jats:tex-math></jats:alternatives></jats:inline-formula> with major axis jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline4" />jats:tex-math$c$</jats:tex-math></jats:alternatives></jats:inline-formula> and minor axis jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline5" />jats:tex-math$b$</jats:tex-math></jats:alternatives></jats:inline-formula> for three different aspect ratios: jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline6" />jats:tex-math$AR=b/c=1$</jats:tex-math></jats:alternatives></jats:inline-formula> (a circle); jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline7" />jats:tex-math$AR=0.5$</jats:tex-math></jats:alternatives></jats:inline-formula>; and jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline8" />jats:tex-math$AR=0.1$</jats:tex-math></jats:alternatives></jats:inline-formula>. The vertical location of these foils jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline9" />jats:tex-math$ys(t)=A\sin (2\pi f0t)$</jats:tex-math></jats:alternatives></jats:inline-formula> oscillates with amplitude jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline10" />jats:tex-math$A$</jats:tex-math></jats:alternatives></jats:inline-formula> and frequency jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline11" />jats:tex-math$f0$</jats:tex-math></jats:alternatives></jats:inline-formula> 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 jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline12" />jats:tex-math$KC=2\pi A/c$</jats:tex-math></jats:alternatives></jats:inline-formula>, and the non-dimensional frequency, or Stokes number jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline13" />jats:tex-math$\beta =f0c2/\nu$</jats:tex-math></jats:alternatives></jats:inline-formula>. We observe three distinct patterns of asymmetry, labelled ‘S-type’ for synchronous asymmetry, ‘jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline14" />jats:tex-math$QPH$</jats:tex-math></jats:alternatives></jats:inline-formula>-type’ and ‘jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline15" />jats:tex-math$QPL$</jats:tex-math></jats:alternatives></jats:inline-formula>-type’ for quasi-periodic asymmetry at sufficiently high and sufficiently low (i.e. jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline16" />jats:tex-math$AR=0.1$</jats:tex-math></jats:alternatives></jats:inline-formula>) aspect ratios, respectively. These patterns are separated at the critical locus in jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline17" />jats:tex-math$KC$</jats:tex-math></jats:alternatives></jats:inline-formula>–jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline18" />jats:tex-math$\beta$</jats:tex-math></jats:alternatives></jats:inline-formula> 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 jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline19" />jats:tex-math$KC$</jats:tex-math></jats:alternatives></jats:inline-formula> as jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline20" />jats:tex-math$AR$</jats:tex-math></jats:alternatives></jats:inline-formula> decreases. We find for the smallest aspect ratio case (jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline21" />jats:tex-math$AR=0.1$</jats:tex-math></jats:alternatives></jats:inline-formula>) that the transition to asymmetry, for all values of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline22" />jats:tex-math$KC$</jats:tex-math></jats:alternatives></jats:inline-formula>, occurs for a critical value of an ‘amplitude’ Stokes number jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline23" />jats:tex-math$\beta A=\beta (KC)2=4\pi 2f0A2/\nu \simeq 3$</jats:tex-math></jats:alternatives></jats:inline-formula>. The jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline24" />jats:tex-math$QPL$</jats:tex-math></jats:alternatives></jats:inline-formula>-type asymmetry for jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline25" />jats:tex-math$AR=0.1$</jats:tex-math></jats:alternatives></jats:inline-formula> is qualitatively different in physical and mathematical structure from the jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline26" />jats:tex-math$QPH$</jats:tex-math></jats:alternatives></jats:inline-formula>-type asymmetry at higher aspect ratio. The flows at the two ends of the ellipse become essentially decoupled from each other for the jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline27" />jats:tex-math$QPL$</jats:tex-math></jats:alternatives></jats:inline-formula>-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 jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline28" />jats:tex-math$QPH$</jats:tex-math></jats:alternatives></jats:inline-formula>-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 jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline29" />jats:tex-math$QPH$</jats:tex-math></jats:alternatives></jats:inline-formula>-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 jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline30" />jats:tex-math$AR=0.1$</jats:tex-math></jats:alternatives></jats:inline-formula>, freedom to move significantly encourages the onset of asymmetry, but the newly observed ‘primary’ jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline31" />jats:tex-math$QPL$</jats:tex-math></jats:alternatives></jats:inline-formula>-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 jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline32" />jats:tex-math$KC$</jats:tex-math></jats:alternatives></jats:inline-formula> and jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112015006618_inline33" />jats:tex-math$\beta$</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>