2.1. Problem description

We consider elliptical foils with major axis $c$ and minor axis $b$ , such that $AR=b/c\leqslant 1$ , with uniform mass density ${\it\rho}_{s}$ , as shown in figure 2(a). The elliptical foil translates in the infinite $x$ – $y$ plane through a two-dimensional viscous fluid of density ${\it\rho}$ and kinematic viscosity ${\it\nu}$ . Although variations in the density ratio ${\it\rho}_{s}/{\it\rho}$ undoubtedly affect the flow dynamics after symmetry breaking (see e.g. Alben & Shelley Reference Alben and Shelley2005), here we are exclusively interested in the initial behaviour very close to the transition boundary, and so for simplicity, we keep the density ratio fixed at the single value ${\it\rho}_{s}/{\it\rho}=10$ .

As noted above, we impose a vertical oscillation of the centre of the foil so that $y_{s}(t)=A\sin (2{\rm\pi}f_{0}t)$ . We generalize the control parameters defined in (1.1) to elliptical foils by using $c$ as the characteristic length, i.e.

Henceforth, all lengths are non-dimensionalized with $c$ , all densities are non-dimensionalized with ${\it\rho}$ , and all times are non-dimensionalized with the viscous time scale $c^{2}/{\it\nu}$ , such that the non-dimensional period $T_{0}$ of the primary oscillation is $1/{\it\beta}$ , and the non-dimensional primary frequency $f_{0}={\it\beta}$ .

Figure 2. (a) Geometric definition of an elliptical foil with $AR=b/c=0.5$ ;(b) representative grid distribution around the foil.

The Stokes number ${\it\beta}$ may thus be thought of as a Reynolds number, involving as it does the balance between inertia and viscosity, or equivalently the relative size of the viscous time scale to the dominant flow time scale, the primary oscillation period of the foil. However, there are alternative definitions using different combinations of characteristic velocities and length scales which may be more relevant. For sufficiently large $KC$ , the flow must depend on the horizontal extent of the ellipse, and so it is natural to define a Reynolds number using the maximum vertical velocity $2{\rm\pi}Af_{0}$ as the velocity scale and the major axis as the length scale,

where the factor of ${\rm\pi}$ makes this definition equivalent to the ‘flapping’ Reynolds number of Vandenberghe et al. (Reference Vandenberghe, Zhang and Childress2004). Typically, for the flows considered here, $Re_{A}\sim O(10{-}100)$ . Conversely, for smaller values of $KC$ , it is at least conceivable that the horizontal extent of the foil does not play a significant dynamical role, and so $A$ should also be used as the natural length scale of the flow, leading to an ‘amplitude’ Stokes number ${\it\beta}_{A}$ defined as

2.2. Numerical method

To simulate the flow around an oscillating foil in a quiescent fluid, we use the open source code OpenFOAM (Jasak Reference Jasak1996). The time-dependent Navier–Stokes equations are solved using the finite volume method, assuming incompressibility. The mass and momentum equations are solved on a moving grid domain using the Arbitrary Lagrangian Eulerian (ALE) formulation (see Ferziger & Peric Reference Ferziger and Peric2002). The integral form of the governing (conservation) equation defined in an arbitrary moving volume $V$ bounded by a closed surface $S$ is:

where $\boldsymbol{U}$ is the fluid velocity, $\boldsymbol{U}_{b}$ is the boundary velocity of a finite volume and $p$ is the pressure. As the volume $V$ is no longer fixed in space, its motion is captured by the motion of its bounding surface $S$ at the boundary velocity $\boldsymbol{U}_{b}$ . For the details of the discretisation and implementation of boundary conditions, as well as the transformation of the underlying partial differential equations into corresponding systems of algebraic equations, see Ferziger & Peric (Reference Ferziger and Peric2002).

The space discretizations are second-order upwind for the convection terms and central differences for the Laplacian terms, respectively. The time discretization is first-order implicit Euler. Pressure–velocity coupling is enforced using the PISO scheme (Ferziger & Peric Reference Ferziger and Peric2002). The preconditioned conjugate gradient (PCG) method is used to treat the pressure equation and preconditioned bi-conjugate gradient (PBiCG) method is used for the velocity equations. Numerical accuracy is set to double precision and the initial conditions are chosen to be uniform. We set the boundary condition on the foil to be moving wall, with no flux normal to the wall. For the class of ‘flying’ oscillation flows, for which the foil is free to move horizontally, the motion of the foil is determined by the horizontal component of the force due to the fluid on the foil, through application of Newton’s second law:

where $x_{b}$ is the horizontal location of the foil, $m_{s}$ is the foil mass given by $m_{s}={\it\rho}_{s}S$ with ${\it\rho}_{s}$ the foil density and $S$ the foil area. The (in general time-varying) horizontal component of the force $F_{x}(t)$ is calculated by integrating the pressure and viscous stresses over the surface of the foil, and then identifying the component in the horizontal $x$ -direction. As already noted, we set the density ratio ${\it\rho}_{s}/{\it\rho}=10$ for all simulations. This ordinary differential equation is solved using a fourth-order Runge–Kutta algorithm.

To assure time-discretization independence, we require two conditions. The first condition is that the Courant number of all cells,  $Co$ , must be less than one, i.e.

where ${\it\delta}t$ is the time step, $|\boldsymbol{U}|$ is the magnitude of the velocity through that cell and ${\it\delta}x$ is the cell width in the direction of the velocity. Note that the maximum $Co$ is usually determined by the smallest cell size, and we continuously ensure that $Co<1$ for all cells. The second condition is that we ensure that there at least 2000 time steps for each primary oscillation period $T_{0}=1/f_{0}$ of oscillation, to ensure that any unsteadiness caused by this oscillation is well resolved. We have found that requiring these two conditions yields time accurate and robust results.

To validate the spatial resolution we use, we have carried out a grid-independence study on a purely oscillating elliptical foil with aspect ratio $AR=1.0$ , i.e. a circular cylinder in two-dimensional space, analogously to the study discussed in detail in Deng, Caulfield & Shao (Reference Deng, Caulfield and Shao2014). We find that meshes with approximately 50 000 cells provide satisfactory and consistent accuracy in space. As an example, we plot the grid near an elliptical foil with $AR=0.5$ in figure 2, which shows the gradual increase of the mesh size from the foil boundary. The domain is defined as a circle with a radius $20c$ . Pressure and all components of the velocity gradient tensor are set to zero on the boundary of the domain. Further confidence in the fidelity of our simulations is gained by the good agreement with previous numerical simulations by Elston et al. (Reference Elston, Sheridan and Blackburn2001) of the calculated transition boundary for flow around an $AR=1.0$ foil shown in figure 1, particularly for low ${\it\beta}$ numbers. We identify the transition boundary between symmetric and asymmetric flow for each aspect ratio analogously to the transition boundary shown in figure 1 by conducting a bisection-like search with different parameter pairs in the numerical simulations to identify close parameter pairs, one of which induces asymmetric flow (marked with a square in the figure) while the other (marked with a triangle) maintains symmetric flow over many oscillation periods of the foil.

3.1. Symmetry breaking for $AR=1.0$

For ‘elliptical’ foils with $AR=1.0$ , i.e. circular cylinders, previous researchers have studied in detail the onset of symmetry breaking for the class of pure vertical oscillation, where the cylinder is held at a fixed horizontal location. As already noted, through the use of a Floquet stability analysis restricted to a two-dimensional subspace, Elston et al. (Reference Elston, Blackburn and Sheridan2006) identified a single marginal stability or transition boundary in $KC$ – ${\it\beta}$ space, but with two distinct asymmetric flow patterns above this curve, both arising from supercritical bifurcations depending on whether the critical Floquet multipliers are real or complex-conjugate pairs. At relatively low values of ${\it\beta}$ , the flows break $x$ -reflection symmetry, while retaining a spatio-temporal symmetry, which manifests itself in the $z$ -vorticity component of the flow field as ${\it\Omega}(x,y,t)=-{\it\Omega}(x,-y,t+T_{0}/2)$ , where $T_{0}$ is the period of oscillation. These flows are synchronous with the oscillatory motion of the cylinder and hence are labelled S-type. A useful diagnostic is the horizontal force time history of the cylinder, i.e. the time variation of the horizontal component of the integral of the pressure and the viscous stresses over the surface of the cylinder. Due to the fact that vortices are shed in both the upward and downward stroke of the foil, synchronous asymmetry is expected to have a periodic structure in the horizontal force dominated by a frequency $f=2f_{0}=2{\it\beta}$ , i.e. twice the frequency of the primary oscillation of the foil.

At higher values of ${\it\beta}>{\it\beta}_{c}$ , ( $(KC_{c},{\it\beta}_{c})$ is the location of the freezing point in parameter space) the flow loses this fundamental synchronization with the cylinder’s motion and a new, secondary, generically incommensurate period arises. Typically, close to the critical value ${\it\beta}_{c}$ , this secondary period is very long, and so may be thought of as coming in from infinity as ${\it\beta}>{\it\beta}_{c}$ . These inherently quasi-periodic flows are labelled QP-type, and Tatsuno & Bearman (Reference Tatsuno and Bearman1990) observed that large vortices of opposite sign are formed in succession for equal numbers of oscillation cycles, direct evidence of the secondary period for such flows.

These two qualitatively different regimes are well-reproduced quantitatively by our numerical simulations, as shown in figure 1. For the QP-type asymmetries, a new longer secondary period $T_{s}$ arises in the horizontal force time history, resulting from the beating between two close frequencies around (twice) the primary oscillation frequency. The synchronous frequency bifurcates into two slightly different frequencies, due to the interaction between successive vortices shedding at relatively high frequency from the oscillating body. When travelling along the transition boundary to lower Stokes numbers, towards the freezing point location $(KC_{c},{\it\beta}_{c})$ , the two bifurcated frequencies gradually approach each other, converging on twice the primary oscillation frequency.

An alternative, and formally equivalent method to determine the location of the freezing point is to consider the ratio of $T_{s}/T_{0}$ , where $T_{0}$ is the period of oscillation of the flapping foil. Elston et al. (Reference Elston, Blackburn and Sheridan2006) derived this ratio from the critical Floquet multipliers, and showed that, as ${\it\beta}\rightarrow {\it\beta}_{c}^{+}$ , $T_{s}/T_{0}\rightarrow \infty$ . However, due to the fact that close to the critical value ${\it\beta}_{c}$ , the secondary period is predicted to be arbitrarily long, it is challenging to determine the freezing point precisely using numerical simulation. Here, we are not concerned with determining the precise location ( $KC_{c},{\it\beta}_{c}$ ) in parameter space defining the freezing point, but rather we wish to investigate the physical properties of the QP-type flows and the S-type flows, and their dependence on aspect ratio for both classes of pure and flying oscillations. For an $AR=1.0$ , purely oscillating cylinder, we estimate that the freezing point lies between $16<{\it\beta}_{c}<28$ .

Figure 3. Twenty-two evenly spaced contours of vorticity between $-60$ (blue) and $+60$ (red) on the asymmetrical side of the transition boundary for a circular purely oscillating foil with $AR=1.0$ when the centre of the foil is at $y=-A$ , its largest magnitude negative displacement for flows with a: (a) QP-type asymmetry with $KC=4.89,{\it\beta}=40$ ; (b) QP-type asymmetry with $KC=5.65,{\it\beta}=28$ ; (c) S-type asymmetry with $KC=6.91$ , ${\it\beta}=16$ ; (d) S-type asymmetry with $KC=8.16,{\it\beta}=12$ .

We consider the flow structures associated with these different types of asymmetric flows around an $AR=1.0$ cylinder in figure 3, where we show instantaneous vorticity contours just on the asymmetrical side of the transition boundary when the cylinder is at its maximum amplitude negative displacement (i.e. at $y_{s}=-A$ ). Panels (a,b) show QP-type asymmetry, while panels (c,d) show S-type asymmetry. Considering the S-type asymmetrical flows first, their structure is entirely consistent with the flow considered by, for example, Tatsuno & Bearman (Reference Tatsuno and Bearman1990). The opposite signed vortices which roll up on either side of the cylinder develop with slightly different strengths, and so as the cylinder reverses in direction, the stronger vortex convects across the cylinder and is shed at an angle relative to the vertical, leading to an induced flow with a broken left–right symmetry, thus reinforcing the fact that one of the vortices is stronger than the other, and so leading to a synchronous asymmetric flow which still retains an up–down symmetry about the equilibrium position of the oscillating cylinder. As is clear from comparison of the two panels (c,d), there is no preference for the direction in which the stronger vortex propagates. Although not shown, the horizontal force $F_{x}(t)$ time history of the cylinder is dominated by an oscillation with twice the frequency of the primary oscillation frequency of the foil.

Figure 4. Computed positions of massless particles advected from close to a circular cylinder, i.e. $AR=1.0$ in flows with parameters (a) $KC=4.89,{\it\beta}=40$ (QP-type asymmetry) after a release during the interval $[94T_{0},126T_{0}]$ ; and (b) $KC=8.16$ , ${\it\beta}=12$ (S-type asymmetry) after a release during the interval $[96T_{0},112T_{0}]$ following the start of pure oscillation with period $T_{0}$ . Blue particles are released continuously from the right of the equator of the circle, and red particles are released continuously from the left of the equator of the circle.

Conversely, for flows with QP-type asymmetry, a secondary period also develops in the flow, distinct from the primary oscillation period, leading to a distinct, at least for flows with parameters far from the freezing point, as shown in panel (a), dipolar structure of large vortices in the far field. Due to transient effects, and the fact that as the freezing point is approached the secondary period can be very long, this structure can be difficult to detect in a single snapshot, such as that shown in panel (b), but there is a qualitative difference between the two types of flow. A particularly instructive way to observe this qualitative difference is to consider the Lagrangian evolution of virtual (massless) particles released close to the oscillating body. We release a continuous stream of particles with velocity $2{\rm\pi}Af_{0}/10$ at a distance $0.1c$ from the ‘equator’ of the oscillating cylinder in a direction $\pm 30^{\circ }$ from the horizontal. We colour the particles blue if they are released on the right of the cylinder, and red if they are released on the left.

To avoid any effect due to initial transients, we show the particle distribution released during the interval $[94T_{0},126T_{0}]$ in figure 4(a) for the QP-type flow shown in figure 3(a) and the particle distribution during the interval $[96T_{0},112T_{0}]$ in figure 4(b) for the S-type flow shown in figure 3(d). For the QP-type flow, the dipolar vortical structures are clearly apparent. The particles shed from either side of the cylinder are initially aligned with the direction of oscillation, but after several cycles they roll up to form large dipolar vortical structures. Large vortices of opposite sign are formed successively during each oscillation cycle, and the arrangement of these vortices is somewhat similar to that in a von Kármán vortex street behind a cylinder in uniform flow, although the sense of rotation of the vortices is opposite to that found in a unidirectional flow wake and the vortices convect themselves orthogonally away from the generating cylinder.

For the S-type flow shown in figure 4(b), the particle distribution is very different, with symmetry about a horizontal line through the equilibrium position of the oscillating cylinder, and the multiple small horseshoe structures clearly being synchronous with the primary oscillation of the cylinder, with no particular larger-scale structure apart from the over-arching curve of the flow asymmetry extending to the right of the cylinder. For both cases it is also clear (and unsurprising) that there is strong communication between the fluid flow either side of the cylinder, due to the inevitable interaction of the large-scale vortices which develop, with the blue and red particles being thoroughly inter-mixed. All these observations accord well with the results presented by Elston et al. (Reference Elston, Blackburn and Sheridan2006).

3.2. Symmetry breaking for $AR=0.5$

The general picture of the transition boundary to asymmetry being characterised by two distinct types of quasi-periodic and synchronous asymmetry for different values of ${\it\beta}$ either side of the freezing point at $(KC_{c},{\it\beta}_{c})$ carries over to flows around elliptical foils with aspect ratios $AR<1$ , although reducing the aspect ratio has an effect on the location of the transition boundary, as shown in figure 5. As for figure 1, the data for which are reproduced here, we conduct a bisection-like search using numerical simulations with different parameter pairs. Parameter pairs marked with triangles maintain symmetric flow over many oscillation periods, while parameter pairs marked with squares induce asymmetric flow. Generically, for a given value of ${\it\beta}$ , the boundary shifts to smaller values of $KC$ as the aspect ratio decreases. Similar values of ${\it\beta}\sim 20$ are associated with the freezing zone value of ${\it\beta}_{c}$ where the symmetry breaking structure switches from QP-type to S-type asymmetry.

Figure 5. Location of transition boundaries as a function of Keulegan–Carpenter parameter $KC$ and Stokes number ${\it\beta}$ as defined in (1.1) for different aspect ratios: $AR=1.0$ (solid line); $AR=0.5$ (dashed line); $AR=0.1$ (dash-dotted line). The marked freezing zone approximates the location of transition between QP-type and S-type asymmetries. Parameter values associated with symmetric flow as obtained by our two-dimensional simulations are marked by ▵, while parameter values associated with asymmetric flow are marked by ▫.

Turning our attention to the flow structures, as shown in figure 6, once again, when the foil is at its largest magnitude negative displacement, although there are clearly differences in detail, the structure for the flows associated with the intermediate aspect ratio elliptical foil with $AR=0.5$ share strong points of similarity with the flows for the circular cylinder shown in figure 3. There is clear evidence of the characteristic dipolar vortical street for the QP-type flow shown in figure 6(a), and the sweeping, synchronous dominant vortex on one side propagating at an angle to the vertical characteristic of S-type flow is apparent in figure 6(d). As noted in the introduction, we refer to this asymmetry as $\text{QP}_{\text{H}}$ -type asymmetry, associated as it is with a sufficiently high-aspect ratio foil so that the QP-type asymmetry is a simple generalization of the previously discussed asymmetry for flow around an $AR=1.0$  foil.

Figure 6. Twenty-two evenly spaced contours of vorticity between $-60$ (blue) and $+60$ (red) on the asymmetrical side of the transition boundary for an elliptical purely oscillating foil with $AR=0.5$ when the centre of the foil is at $y=-A$ , its largest magnitude negative displacement for flows with a: (a) $\text{QP}_{\text{H}}$ -type asymmetry with $KC=3.01,{\it\beta}=60$ ; (b) $\text{QP}_{\text{H}}$ -type asymmetry with $KC=4.39,{\it\beta}=19$ ; (c) S-type asymmetry with $KC=5.65,{\it\beta}=11$ ; (d) S-type asymmetry with $KC=8.16,{\it\beta}=5$ .

This resemblance is further confirmed by the Lagrangian massless particle distributions, (for the $\text{QP}_{\text{H}}$ -type flow shown in figure 6 a and the S-type flow shown in figure 6 d) which we present in figure 7. As before, the particles are introduced after any initial transient effects have passed. The dipolar vortex street for the $\text{QP}_{\text{H}}$ -type flow for the foil with $AR=0.5$ is really rather similar to the equivalent flow around the circular cylinder shown in figure 4 and the synchronous S-type flow also exhibits a symmetry about a horizontal line through the equilibrium position of the foil, with identifiable structures associated with each of the oscillation periods, although the characteristic curved structure actually now bends round at the end, indicative of a return flow towards the foil. The dipolar vortex street arises from interactions between vortices shed from either side of the foil during both the up-stroke and the down-stroke of the foil, and so we expect to be able to detect a frequency close to $2f_{0}=2{\it\beta}$ in the horizontal force history of the foil.

Figure 7. Computed positions of massless particles advected from close to the elliptical foil with $AR=0.5$ in flows with parameters (a) $KC=4.89,{\it\beta}=19$ ( $\text{QP}_{\text{H}}$ -type asymmetry) after a release during the interval $[130T_{0},150T_{0}]$ ; and (b) $KC=8.16$ , ${\it\beta}=5$ (S-type asymmetry) after a release during the interval $[130T_{0},170T_{0}]$ following the start of pure oscillation with period $T_{0}$ . Blue particles are released continuously from the right of the equator of the circle, and red particles are released continuously from the left of the equator of the circle.

Figure 8. Time dependence of the horizontal force $F_{x}(t)$ on a purely oscillating elliptical foil with aspect ratio $AR=0.5$ at (a) $KC=3.01,{\it\beta}=60$ , representing a typical $\text{QP}_{\text{H}}$ -type asymmetry, and (b) $KC=8.16,{\it\beta}=5$ , representing a typical S-type asymmetry. Time is scaled with the period of the primary oscillation $T_{0}=1/{\it\beta}$ .

The identification of these flows as being of the generic $\text{QP}_{\text{H}}$ -type and S-type asymmetries can be further confirmed by considering the time-dependent properties of the non-dimensional horizontal force histories $F_{x}(t)$ for these two flows, which we plot in figure 8(a) for the $\text{QP}_{\text{H}}$ -type flow with $KC=3.01,{\it\beta}=60$ and in figure 8(b) for the S-type flow with $KC=8.16,{\it\beta}=5$ . The different spectral properties of the two flows are immediately apparent, with the S-type flow being dominated by (twice) the primary frequency of oscillation of the foil, while there is a clear beating of the force signal for the $\text{QP}_{\text{H}}$ -type flow over a much longer period, although there is also a dependence quite close to twice the primary frequency of oscillation. Unsurprisingly, the magnitude (non-dimensionalized with ${\it\nu}$ and $c$ ) of the force is substantially larger for the $\text{QP}_{\text{H}}$ -type flow than for the S-type flow.

These observations can be made quantitative by consideration of the frequency power spectra obtained by using a fast Fourier transform, which we plot in figure 9. It is important to remember that, because of the up–down symmetry of the primary oscillating frequency of the foil, the frequency inherent in horizontal force time history for a symmetric flow should be precisely double that of the primary oscillation frequency $f_{0}={\it\beta}$ of the foil. We observe that two similar, but different, frequencies exist for the $\text{QP}_{\text{H}}$ -type flow asymmetry corresponding to $1.88f_{0}$ and $f_{2}=2.12f_{0}$ . Naturally, there is also substantial power in another lower frequency $f_{3}=0.12f_{0}$ that is exactly the consequence of beating between the other two frequencies. As expected, the horizontal force time history for the S-type flow is completely dominated by twice the primary frequency of oscillation, with a weak contribution of the second harmonic.

Figure 9. Horizontal force power spectra for the flow with a purely oscillating elliptical foil with $AR=0.5$ and (a) $KC=3.01,{\it\beta}=60$ ; (b) $KC=8.16,{\it\beta}=5$ . Frequencies are scaled with the non-dimensional frequency of the primary oscillation $f_{0}={\it\beta}$ .

3.3. Symmetry breaking for $AR=0.1$

The properties of the transition boundary, and indeed the properties of the associated asymmetric flows, are qualitatively different for pure oscillation of the elliptical foil with the smallest aspect ratio $AR=0.1$ , where the foil is, in some sense, long and thin. Consideration of figure 5 suggests that there is a significant change in the structure of the transition boundary curve for the smallest value of $AR=0.1$ . As noted above during the discussion of the different possible definitions for Reynolds number, it is at least plausible for such a flow to be dynamically unaffected by the horizontal extent $c$ of the foil, and so in figure 10, we replot the transition boundaries shown in figure 5 using ${\it\beta}_{A}={\it\beta}(KC)^{2}$ as defined in (2.3) as the $y$ -coordinate to test the hypothesis that $c$ is not significant for the foil with aspect ratio $AR=0.1$ . This hypothesis proves to be correct, as the transition boundary for the $AR=0.1$ case occurs, to a very good approximation, at a fixed value of ${\it\beta}_{A}\simeq 3$ for all calculated values of $KC$ , suggesting that the dependence on $c$ is not significant for this case.

Figure 10. Location of transition boundaries as a function of Keulegan–Carpenter parameter $KC$ and amplitude Stokes number ${\it\beta}_{A}={\it\beta}(KC)^{2}$ as defined in (2.3) for different aspect ratios, $AR=1.0$ (solid line); $AR=0.5$ (dashed line); $AR=0.1$ (dash-dotted line). The marked freezing zone approximates the location of transition between QP-type and S-type asymmetries. Parameter values associated with symmetric flow as obtained by our two-dimensional simulations are marked by ▵, while parameter values associated with asymmetric flow are marked by ▫.

Figure 11. Twenty-two evenly spaced contours of vorticity between $-100$ (blue) and $+100$ (red) on the asymmetrical side of the transition boundary for an elliptical purely oscillating foil with $AR=0.1$ when the centre of the foil is at $y=-A$ , its largest magnitude negative displacement for flows with a: (a) $\text{QP}_{\text{L}}$ -type asymmetry with $KC=1.51,{\it\beta}=60$ ; (b) $\text{QP}_{\text{L}}$ -type asymmetry with $KC=1.88,{\it\beta}=34$ ; (c) S-type asymmetry with $KC=3.14,{\it\beta}=14$ ; (d) S-type asymmetry with $KC=4.39,{\it\beta}=8$ .

Consistent evidence that the small aspect ratio flow is different can also be gained from consideration of the vorticity contours shown in figure 11 for the flow around the elliptical foil with $AR=0.1$ on the asymmetrical side of the transition boundary for $\text{QP}_{\text{L}}$ -type flows, i.e. quasi-periodic asymmetrical flows around sufficiently low aspect-ratio foils (panels a,b) and S-type flows (panels c,d). Considering the S-type flows first, by comparison with equivalent figures 3 and 6 for the other two foils, the induced vortices appear not to be as elongated, and are more strongly localized in the vicinity of the oscillating foil. However, the dynamics is still synchronous and the time history of the horizontal force is still dominated completely by the expected frequency, twice the primary frequency of the oscillating foil i.e. $2f_{0}=2{\it\beta}$ .

Figure 12. Computed positions of massless particles advected from close to the elliptical foil with $AR=0.1$ in flows with parameters (a) $KC=1.51,{\it\beta}=60$ ( $\text{QP}_{\text{L}}$ -type asymmetry) after a release during the interval $[540T_{0},640T_{0}]$ ; and (b) $KC=4.39,{\it\beta}=8$ (S-type asymmetry) after a release during the interval $[130T_{0},170T_{0}]$ following the start of pure oscillation with period $T_{0}$ . Blue particles are released continuously from the right of the equator of the circle, and red particles are released continuously from the left of the equator of the circle.

There is a much more marked and qualitative difference in the structure of the $\text{QP}_{\text{L}}$ -type asymmetry shown in panels (a,b). Rather than a dipolar structure in the far field, the vorticity structure appears appreciably more complex, with a very marked asymmetry between the flow either side of the foil, suggesting that the structure of this $\text{QP}_{\text{L}}$ -type flow is markedly different for foils with low aspect ratios. The lack of observed interaction between the two sides suggests that the up and down strokes of the foil may well play distinct and different roles in the development of asymmetry within this flow, which would imply that the characteristic close frequencies which lead to the observed quasi-periodicity should be centred around the primary frequency of oscillation $f_{0}={\it\beta}$ rather than its first harmonic $2f_{0}$ , as observed for the foils with larger aspect ratio.

This suggestion is confirmed by consideration of the Lagrangian evolution of massless particles for the flows shown in figure 11(a,d). We plot in figure 12(a,b) the particle distributions after release once the flow is in quasi-steady state during the time intervals $[540T_{0},640T_{0}]$ and $[60T_{0},88T_{0}]$ , respectively. Both these patterns are markedly different from the equivalent patterns for the larger aspect-ratio foils shown in figures 4 and 7. The S-type pattern is still synchronous but is, as expected, much more localized in the vicinity of the foil, and strongly asymmetric between the two ends of the elliptical foil. The distribution of particles is also much wider, indicative of the different aspect ratios of the dominant vortices which develop around the flapping foil.

The difference for the $\text{QP}_{\text{L}}$ -type asymmetry is once again more marked, and the flow is so different that we believe that this is a distinct new type of quasi-periodic asymmetry. There is much less evidence of a dipolar vortex street developing away from the oscillating foil, but even more noticeable is the marked left–right asymmetry. Unlike all the other cases considered, there is virtually no communication between the fluid in the vicinity of either side of the equator of the flapping foil. Virtually all the blue massless particles released to the right of the foil remain there, and only the red particles released on the left of the foil are propagated any significant distance away from the oscillating foil, though at a substantially shallower angle than the (essentially vertical) propagation observed for the flows induced by the other two aspect ratio foils.

Figure 13. Pressure profiles over the foil surface at the time-instant corresponding respectively to the $\text{QP}_{\text{L}}$ -type flow shown in figure 11(a) (solid line, normalized by its maximum value 2.272), and the S-type flow shown in figure 11(d) (dashed line, normalized by its maximum value 0.138).

The different physical response is also evident in the normalized integrated pressure distribution over the surface of the foils plotted in figure 13 for the $\text{QP}_{\text{L}}$ -type flow with $KC=1.51,{\it\beta}=60$ (solid line) and the S-type flow with $KC=4.39,{\it\beta}=8$ (dashed line) at the same time instants, as shown in figure 11. The pressure distribution is both more closely (though not completely) symmetric and substantially higher in amplitude for the $\text{QP}_{\text{L}}$ -type asymmetry flow. Interestingly, and perhaps counter-intuitively, this relatively symmetric and strong pressure distribution suppresses the communication between the induced vortices at either tip of the foil, thus leading to the marked separation between the evolving dynamics at either tip, as is particularly evident in the massless particle distributions shown in figure 12(a). Since there is virtually no coupling between the vortices at either tip, the symmetry breaking leads to one set of massless particles (those on the left) being advected away from the foil due to the broken symmetry. Conversely, the small in magnitude, yet highly asymmetric pressure distribution evident for the flow with S-type asymmetry leads to the curling over the foil of the slightly stronger (negative) vortex shed from the left tip of the foil apparent in figure 11(d), which interacts strongly with the positive vortex at the right tip of the foil, thus leading to the distinctive intermingled particle distribution curving over the foil shown in figure 12(b).

Figure 14. Time dependence of the horizontal force on the purely oscillating elliptical foil with aspect ratio $AR=0.1$ at $KC=1.51,{\it\beta}=60$ , representing QP-type asymmetry. Time is scaled with the period of the primary oscillation $T_{0}=1/{\it\beta}$ .

The qualitatively different, and yet still inherently quasi-periodic, character of the $\text{QP}_{\text{L}}$ -type asymmetric flow is confirmed by consideration of the spectral properties of the time history of the horizontal force on the foil. For the flow shown in figure 11(a), we plot the time history of the horizontal force on the foil in figure 14. The time history shows an extremely long transient behaviour, although quasi-periodic beating is apparent from very early on in the flow evolution. Eventually, a clear, yet long secondary period $T_{s}\simeq 23T$ emerges. We analyse the spectral properties of the horizontal force once this periodic oscillation saturates, and the flow reaches a quasi-steady state, calculating the power spectrum for the horizontal force time history over the interval $[600T_{0},900T_{0}]$ as plotted in figure 15.

Figure 15. Horizontal force power spectrum for the flow with a purely oscillating elliptical foil with $AR=0.1$ and $KC=1.51,{\it\beta}=60$ with $\text{QP}_{\text{L}}$ -type asymmetry for the time interval $[600T_{0},900T_{0}]$ . Frequencies are scaled with the non-dimensional frequency of the primary oscillation $f_{0}={\it\beta}=1/T_{0}$ .

The power spectrum is qualitatively different from the power spectrum for the $\text{QP}_{\text{H}}$ -type asymmetric flow for an elliptical foil with $AR=0.5$ , as plotted in figure 9(a). There still remains an (unsplit) component at twice the primary oscillation frequency $2f_{0}$ , but quasi-periodicity is here due to two closely separated frequencies either side of the primary oscillation frequency itself, i.e. either side of $f_{0}={\it\beta}$ . This is consistent with the lack of communication between the two ends of the foil seen in the vortical contours and the Lagrangian massless particle distributions. The fact that vortex interaction leading to quasi-periodicity appears to occur only on one side implies that the dominant quasi-periodic frequencies will be close to the primary oscillation frequency $f_{0}$ , not its harmonic $2f_{0}$ (which would be characteristic of tip–tip interaction), exactly as observed. Therefore, as already mentioned in the introduction, we refer to this type of asymmetry as a ‘primary’ $\text{QP}_{\text{L}}$ -type asymmetry.

3.4. Mathematical analysis of $\text{QP}_{\text{H}}$ -type and $\text{QP}_{\text{L}}$ -type asymmetric flows

Further evidence that there are two distinct types of quasi-periodic asymmetries depending on the aspect ratio of the foil can be gained by considering quantitatively the Floquet stability of the oscillating base (symmetric) flows. Floquet stability analysis examines the behaviour of a perturbation, $u^{\prime }$ , compared to a $T$ -periodic (limit cycle) base flow, $U$ , to determine whether the perturbation grows or decays from cycle to cycle. The evolution equations for the perturbation flow are the Navier–Stokes equations linearized about the base, (in this case symmetric) flow around the foil. Perturbation solutions can be written as a sum of components $\tilde{u} (t_{0})\text{e}^{{\it\sigma}(t-t_{0})}$ where $\tilde{u} (t_{0})$ is a $T$ -periodic Floquet eigenfunction, evaluated at arbitrary phase $t_{0}$ and ${\it\sigma}$ is a Floquet exponent. In general, a Floquet multiplier is defined by ${\it\mu}=\text{e}^{{\it\sigma}T}$ where $T$ is the period of the base periodic flow. The exponents ${\it\sigma}$ and the multipliers ${\it\mu}$ can either be real, or occur in complex-conjugate pairs. Instability occurs when a multiplier leaves the unit circle, $|{\it\mu}|>1$ , or equivalently when the real part of a Floquet exponent becomes positive.

The technique we use for Floquet stability analysis is a Krylov subspace method that examines the stability of the linearized Poincaré map for the perturbation flow, and is detailed in Elston et al. (Reference Elston, Sheridan and Blackburn2004, Reference Elston, Blackburn and Sheridan2006) who, as already mentioned, considered the Floquet stability of the flow around a circle with $AR=1$ . The reflection symmetry of the base flow is enforced by solving in a half domain (see figure 2), with symmetry boundary conditions along the $x=0$ boundary. The base flow is integrated in time for 30 cycles, when it reaches a periodic state. It is then projected onto the full domain, and stored for Fourier time interpolation. We store 64 time slices, equi-spaced in time over the base flow period $T$ , for reconstruction of the base flow. It should be noted that for two-dimensional Floquet analysis in the current problem, there is difficulty resolving stable modes, $|{\it\mu}|<1$ , while unstable modes are resolved without difficulty, and the location of marginal stability can be estimated by extrapolation to $|{\it\mu}|=1$ (Elston et al. Reference Elston, Sheridan and Blackburn2004).

Figure 16. Floquet results at ${\it\beta}=60$ for $AR=1.0$ . (a) The variation of $|{\it\mu}|$ with $KC$ : solid circles represent multipliers for the mode of marginal stability, while open circles show magnitude for a complex-conjugate pair of multipliers that cross the unit circle at $KC=4.114$ . (b) The phase angle for the complex-conjugate pair of multipliers. (c) The secondary periods computed from $T_{s}=2{\rm\pi}T/{\it\theta}$ , with a dashed line showing inverse square-root behaviour ( $T_{s}/T=1.633+3.362/\sqrt{4.319-KC}$ ).

Figure 17. Floquet results at ${\it\beta}=60$ for $AR=0.5$ . (a) The variation of $|{\it\mu}|$ with $KC$ : solid circles represent multipliers for the mode of marginal stability, while open circles show magnitude for a complex-conjugate pair of multipliers that cross the unit circle at $KC=2.751$ . (b) The phase angle for the complex-conjugate pair of multipliers. (c) The secondary periods computed from $T_{s}\simeq 2{\rm\pi}T/{\it\theta}$ , with a dashed line showing inverse square-root behaviour ( $T_{s}/T=4.546+4.245/\sqrt{3.121-KC}$ ).

Figure 18. Floquet results at ${\it\beta}=60$ for $AR=0.1$ . Note that all calculated Floquet multipliers are real.

As pointed out by Elston et al. (Reference Elston, Sheridan and Blackburn2004) for an oscillating circular cylinder at sufficiently high ${\it\beta}$ , where the symmetry breaking is observed to be of QP-type, the first multipliers to cross the unit circle occur in complex-conjugate pairs, i.e. ${\it\mu}=\text{e}^{\pm \text{i}{\it\theta}}$ , so the (supercritical) bifurcation is of Neimark–Sacker type. We plot the Floquet results at ${\it\beta}=60$ for $AR=1.0$ , $AR=0.5$ and $AR=0.1$ in figures 16–18 respectively. In Elston et al. (Reference Elston, Sheridan and Blackburn2004) they showed by comparison to two-dimensional direct numerical simulation results that the simple relationship $T_{s}/T\simeq 2{\rm\pi}/{\it\theta}$ held very close to transition for the specific value of ${\it\beta}=44.2$ . As is apparent from careful consideration of figures 3 and 9 of Elston et al. (Reference Elston, Sheridan and Blackburn2004), that particular choice of ${\it\beta}$ shows excellent agreement between the results of the numerical simulations and Floquet analysis, whereas there is some relatively small quantitative difference between the Floquet analysis and the numerical simulations at ${\it\beta}=60$ . We observe a similar slight numerical mismatch, with the critical value of $KC$ predicted by Floquet analysis for $AR=1$ being $KC_{c}=4.114$ , compared to the numerical simulation indicating that $KC_{c}\in [4.27,4.39]$ and for $AR=0.5$ being $KC_{c}=2.751$ , compared to the numerical simulation indicating that $KC_{c}\in [2.89,3.01]$ .

Figures 16(c) and 17(c) show the ratios $T_{s}/T$ derived from the assumed correspondence (as presented by Elston et al. (Reference Elston, Sheridan and Blackburn2004, Reference Elston, Blackburn and Sheridan2006)) with the Floquet multipliers calculated for $AR=1.0$ and $AR=0.5$ , respectively. For both flows, as $KC\rightarrow KC_{\infty }$ , ( $KC_{\infty }=4.319$ for $AR=1$ and $KC_{\infty }=3.121$ for $AR=0.5$ ) $T_{s}/T\sim \sqrt{KC_{\infty }-KC}\rightarrow \infty$ . Such a square-root scaling is indicative of a ‘saddle node on an invariant circle’ or ‘SNIC’ bifurcation (see e.g. Lopez, Rubio & Marques (Reference Lopez, Rubio and Marques2006), Rubio, Lopez & MArques (Reference Rubio, Lopez and MArques2008) for a detailed discussion) although care must be taken in drawing any conclusions from the inherently linear and two-dimensional Floquet stability analysis for values of $KC$ far, in some sense, from the transition boundary to asymmetry. However, it is at least plausible that as ${\it\beta}$ is reduced along the transition boundary, with $KC$ increasing towards the freezing zone where the $\text{QP}_{\text{H}}$ -type asymmetry disappears as $T_{s}\rightarrow \infty$ , this divergence in the secondary period will also be consistent with a SNIC bifurcation.

Irrespective of this speculation, the Floquet stability results for $AR=0.1$ show a totally different behaviour compared to the higher aspect ratio counterparts, i.e. $AR=1.0$ and $AR=0.5$ . As shown in figure 18, the first Floquet multiplier to cross the unit circle occurs with a single real value. Though we have clearly shown in figures 12 and 14 that a secondary period emerges for $AR=0.1$ in the higher ${\it\beta}$ region, it appears that this quasi-periodic behaviour for low aspect ratios cannot be predicted by a conventional linear Floquet stability analysis.

Furthermore, consideration of phase portraits and appropriate Poincaré maps based around iterates of the horizontal force and its derivatives demonstrate qualitative differences between the $\text{QP}_{\text{H}}$ -type asymmetry and the $\text{QP}_{\text{L}}$ -type asymmetry. In figure 19(a,c) we plot a phase portrait of the dynamics during time intervals when the flow has settled into quasi-periodic behaviour, showing the variation of $\text{d}F_{x}/\text{d}t$ against $F_{x}$ for the $\text{QP}_{\text{H}}$ -type asymmetry in the flow with $AR=0.5$ , $KC=3.01$ , ${\it\beta}=60$ and the $\text{QP}_{\text{L}}$ -type asymmetry in the flow with $AR=0.1$ , $KC=1.51$ , ${\it\beta}=60$ . The $\text{QP}_{\text{H}}$ -type asymmetry in figure 19(a) shows a relatively simple quasi-periodicity involving two similar loops, with relatively slow rate of change of $F_{x}$ occurring when $F_{x}\simeq \pm (1{-}1.5)$ , interspersed with much faster variation away from these points. This picture is reinforced by the Poincaré map shown in figure 19(b). Once per primary period we plot $F_{x}$ against its value five primary periods earlier, and there is a collection of points in the vicinity of $F_{x}\simeq \pm (1{-}1.5)$ , reinforcing the slow–fast character of this quasi-periodic asymmetry, and its connection to a SNIC bifurcation (cf. figure 11 of Rubio et al. (Reference Rubio, Lopez and MArques2008)).

On the other hand, the phase portrait and Poincaré map shown in figure 19(b,d) for the flow with $AR=0.1$ , $KC=1.51$ , ${\it\beta}=60$ exhibiting $\text{QP}_{\text{L}}$ -type asymmetry is qualitatively different, and much more complex. Both figures show that the horizontal force wanders over phase space, with a more significant lower frequency component than the flow exhibiting $\text{QP}_{\text{H}}$ -type asymmetry, and in particular there is no real evidence of any slow–fast dynamics.

Figure 19. Phase portraits plotting the time rate of change $\text{d}F_{x}/\text{d}t$ of the horizontal force against $F_{x}$ for a flow with: (a) $AR=0.5$ , $KC=3.01$ , ${\it\beta}=60$ as shown in figure 8(a) exhibiting $\text{QP}_{\text{H}}$ -type asymmetry over the time interval $t/T_{0}=60{-}100$ ; (c) $AR=0.1$ , $KC=1.51$ , ${\it\beta}=60$ as shown in figure 14 exhibiting $\text{QP}_{\text{L}}$ -type asymmetry over the time interval $t/T_{0}=400{-}540$ , covering approximately 6 secondary periods in each case. Panels (b) and (d) are reconstructed phase portraits using delays of five primary periods, for the flows corresponding to (a) and (c) respectively. For clarity, a single iterate per primary period is plotted on the Poincaré map.

Figure 20. Location of transition boundaries as a function of Keulegan–Carpenter parameter $KC$ and Stokes number ${\it\beta}$ as defined in (1.1) for flows associated with purely oscillating foils (solid lines) and flying oscillating foils (dashed lines) with: (a) aspect ratio $AR=1.0$ ; (b) aspect ratio $AR=0.5$ ; (c) aspect ratio $AR=0.1$ . (d) Location of transition boundaries as a function of Keulegan–Carpenter parameter $KC$ and Stokes number ${\it\beta}_{A}$ as defined in (2.3) for flows associated with purely oscillating foils (solid lines) and flying oscillating foils (dashed lines) with aspect ratio $AR=0.1$ . Parameter values associated with symmetric flow as obtained by our two-dimensional simulations are marked by small filled squares for pure oscillations and by large open squares for flying oscillations, while parameter values associated with asymmetric flow are marked by small filled circles for pure oscillations and by large open circles for flying oscillations.

In conclusion, we have identified what we believe to be a new ‘primary’ $\text{QP}_{\text{L}}$ -type of quasi-periodic asymmetric flow for small aspect ratio foils undergoing ‘pure’ oscillation, which is associated with the primary frequency of oscillation in the horizontal force time history. It is clear that an extensive parametric study would be needed to clarify all aspects of the transition to this $\text{QP}_{\text{L}}$ -type of asymmetry from the previously identified $\text{QP}_{\text{H}}$ -type asymmetry, a simple generalization of the QP-type asymmetry for circular cylinders to moderate aspect ratios, associated with twice the primary frequency of oscillation in the horizontal force time history. Such a study is beyond the scope of this paper, where we now turn our attention to consideration of how the transition boundary properties are modified for the class of flying oscillations where the foil is free to move horizontally. We are particularly interested in investigating whether these two different QP-type asymmetries occur for foils which are allowed to move horizontally, as it is not clear whether the quasi-periodicity growth mechanisms rely inherently on the foil being fixed horizontally.

4.1. Suppression of synchronous asymmetry for flying oscillations

As shown in figure 20(a), for flow round a circular cylinder with $KC=6.91,{\it\beta}=16$ , synchronous S-type asymmetry develops for pure oscillation, while symmetry actually continues to persist at the same point in parameter space for flying oscillation. The vorticity contours for the pure oscillation, also shown in figure 3(a), are replotted close to the cylinder in figure 21(a) for comparison with the vorticity contours from the flying oscillation shown in figure 21(b), at the same instant in the flow evolution when the oscillating cylinder is at its greatest (negative) displacement.

There is a marked difference between the two flow structures, which is also consistent with the difference of the horizontal force on the oscillating cylinder, as shown in figure 22 where the force is three orders of magnitude larger for the pure oscillation flow than for the flying oscillation flow. Spectral analysis of these time histories, as shown in figure 23, shows a very strong synchronous frequency (at twice the primary frequency) for the flow with pure oscillation, indicative of the strong asymmetry visible in the flow. There is an appreciably weaker synchronous frequency also apparent in the power spectrum for the flying oscillation case shown in figure 21(b), and there is some very weak asymmetry apparent in the vorticity field shown in figure 21(b). Therefore, it seems appropriate to state that flying oscillations suppress rather than eliminate synchronous asymmetry near the transition boundary on the S-type asymmetry side of the freezing point.

Figure 22. Time dependence of the horizontal force on a circular cylinder with parameters $AR=1.0$ , $KC=6.91$ and ${\it\beta}=16$ , subject to (a) a pure oscillation and (b) a flying oscillation. Time is scaled with the period of the primary oscillation $T_{0}=1/{\it\beta}$ . Note the relative size of the vertical axes.

Figure 23. Horizontal force power spectra for the flow with a circular cylinder with parameters $AR=1.0$ , $KC=6.91$ and ${\it\beta}=16$ subject to (a) a pure oscillation and (b) a flying oscillation, corresponding to figure 22(a,b) respectively. Frequencies are scaled with the non-dimensional frequency of the primary oscillation $f_{0}={\it\beta}$ .

To gain physical insight into how this suppression occurs, we consider another pair of flows close to the S-type asymmetry transition boundary, where the purely oscillating foil is associated with a strong synchronous asymmetric flow, while the flying oscillating foil is associated with a (very close to) symmetric flow. As shown in figure 20(b), particular flow parameters with the required properties are those for an elliptical foil with $AR=0.5$ , $KC=8.16$ and ${\it\beta}=5$ , and the purely oscillating flow has been discussed in detail in § 3.2, and aspects of this flow are shown in figures 6(d), 7(b), 8(b) and 9(b). Here, we impose a pure oscillation so that the foil has a fixed horizontal location until it converges to an essentially steady state. We assess this convergence by considering the time history of the horizontal force on the foil, as shown in figure 24. The envelope of the horizontal force grows, and then saturates, after approximately 70 primary oscillation periods. At this stage, the synchronous asymmetry in the flow is well established, and there is a clear bias towards a negative horizontal force on the elliptical foil. We then release the fixed horizontal location of the foil, allowing it to ‘fly’ after 80 primary oscillation periods. The effect on the horizontal force is dramatic, and instantaneous, as it drops very rapidly, and essentially exponentially, and after a slight overshoot relaxes back to extremely small values, of the order of the initial numerical fluctuations.

Figure 24. Time dependence of the horizontal force on an elliptical foil with parameters $AR=0.5$ , $KC=8.16$ and ${\it\beta}=5$ , for an initially ‘pure’ oscillation for $0\leqslant t/T_{0}\leqslant 80$ which is then released to execute flying oscillations for $t>80T_{0}$ , where $T_{0}=1/{\it\beta}$ is the period of the primary oscillation.

Figure 25. (a–k) Twenty-two evenly spaced contours of vorticity between $-60$ (blue) and $+60$ (red) at evenly spaced time intervals during the first primary period after the foil is released to ‘fly’ for the flow around an elliptical foil with parameters $AR=0.5$ , $KC=8.16$ and ${\it\beta}=5$ ; (l) 22 evenly spaced contours of vorticity between $-60$ (blue) and $+60$ (red) for the flow five periods after the release of the foil to move horizontally.

The mechanism by which this rapid return to (close to) symmetric flow occurs can be understood by considering the vorticity contours shown in figure 25(a–k), which show the vorticity at 11 evenly spaced time intervals directly after the foil is released in panels. The left–right asymmetry associated with this flow is clearly apparent. On both the upward and downward strokes the induced vortex to the left of the foil dominates the vortex to the right, in that it is more elongated, slightly stronger and angled towards the right, leading to the strong (and tilted to the right) asymmetry observed in the Lagrangian particle distributions shown in figure 7(b). Significantly, as the slightly stronger vortex is shed from the left side of the foil, it imposes a slightly weaker force on the foil than the vortex to the right, since it induces a slightly lower pressure on the left. Since the foil is now free to move, the foil moves to the left. This is in response to the negative (on average) horizontal forces, shown in figure 24. This leftward motion has two key components. First, it tends to weaken the subsequent vortex generation to the left, and strengthen the vortex generation to the right, thus exerting a stabilizing influence on the asymmetry growth mechanism. Second, and very importantly, this stabilizing influence is in turn synchronous with the primary oscillation of the foil and with the asymmetry, and so is precisely tuned to symmetrise the flow, as is apparent in figure 25(l), where the vorticity contours are plotted 5 periods after the foil is released. This leftward motion of the foil actually ‘overshoots’ before the symmetry of the flow has been completely restored, leading to the weak net positive horizontal force on the foil, which causes the foil ultimately to move (very slowly) rightwards with a velocity approximately 0.1 % of the maximum vertical velocity associated with the primary oscillation. This slight drift appears to control the flow very close to symmetry for long periods.

4.2. Encouragement of quasi-periodic asymmetry for flying oscillations

For the two larger aspect ratios which we have considered, the situation is qualitatively different for flows with parameters to the left of the freezing point, where the asymmetry, when it onsets, is of $\text{QP}_{\text{H}}$ -type. In this region of parameter space, by consideration of figure 20, allowing the oscillating foil to move horizontally, actually encourages the onset of quasi-periodic asymmetry. To investigate the physical mechanisms underlying this phenomenon, we consider the flow around an elliptical foil with $AR=0.5$ , with ${\it\beta}=40$ . As is shown in figure 20(b), when $KC=3.2$ , if the elliptical foil is free to move horizontally, we observe high-aspect ratio quasi-periodic $\text{QP}_{\text{H}}$ -type asymmetry. However, if we fix the horizontal location of the foil, the flow rapidly returns to symmetry, and $\text{QP}_{\text{H}}$ -type asymmetry only arises when $KC$ is increased to 3.32.

Figure 26. Twenty-two evenly spaced contours of vorticity between $-60$ (blue) and $+60$ (red) for flow around an elliptical foil with aspect ratio $AR=0.5$ when the foil is at its minimum vertical location for flows with parameters $KC=3.2,{\it\beta}=40$ subject to (a) a pure oscillation and (b) a flying oscillation.

In figure 26(a), we plot vorticity contours around the purely oscillating foil with aspect ratio $AR=0.5$ , when it is at its minimum vertical location, for a flow with ${\it\beta}=40$ and $KC=3.2$ . The contours are close to symmetric, unlike the equivalent contours for the flying oscillation shown in figure 26(b), which exhibit a marked $\text{QP}_{\text{H}}$ -type asymmetry. The time histories of the corresponding horizontal forces for these pure and flying oscillations are shown in figure 27(a,b), respectively. Unsurprisingly, the forces associated with the pure oscillation remain at very low values throughout the whole time history, although there is some complicated spectral structure manifest as beating of the very small-amplitude signal, while for the flying oscillation, the horizontal force increases rapidly after an initial transient to large quasi-periodic values, implying a significant symmetry breaking process eventually arising with no specific forcing (apart from numerical round-off error).

Figure 27. Time dependence of the horizontal force on an elliptical foil with aspect ratio $AR=0.5$ in a flow with parameters $KC=3.2,{\it\beta}=40$ subject to (a) a pure oscillation and (b) a flying oscillation. Time is scaled with the primary oscillation period $T_{0}=1/f_{0}=1/{\it\beta}$ . Note the relative size of the vertical axes.

Figure 28. Horizontal force power spectra for the flow around an elliptical foil with aspect ratio $AR=0.5$ with parameters $KC=3.2$ and ${\it\beta}=40$ subject to: (a) a pure oscillation; and (b) a flying oscillation, corresponding to figure 27(a,b) respectively. Frequencies are scaled with the non-dimensional frequency of the primary oscillation $f_{0}={\it\beta}$ .

Considering the spectral properties of these horizontal force time histories, as plotted in figure 28, it is clear the ‘symmetric’ flow does indeed contain frequencies either side of the primary oscillation frequency, leading to a further, much smaller, beating frequency. Nevertheless, the primary frequency is still strongly dominant, and it is appropriate to consider the flow to be (at least close to) symmetric. On the other hand, there is clear evidence of dominant frequency splitting either side of (twice) the primary oscillation frequency for the flying oscillation shown in figure 28(b), with a strong long secondary period, as is also evident in the time history of the horizontal force plotted in figure 27(b). This is entirely consistent with the $\text{QP}_{\text{H}}$ -type asymmetry for purely oscillating foils with sufficiently high aspect ratio as discussed above.

The physical interpretation of this encouragement of the onset of $\text{QP}_{\text{H}}$ -type asymmetry for flying foils has several aspects. At high ${\it\beta}$ , the period of the primary oscillation is short, and so it is possible for complex interactions between the vortices associated with different primary oscillations to occur. If the foil is free to move, these interactions can lead to the foil being displaced from its equilibrium position(J. Zhang, 2012, private communication). Furthermore, such a displacement can encourage the development of slight perturbations to the frequencies with which the vortices are shed, leading to the possibility of the development of a long secondary period through beating between two close, but different, frequencies of the induced flow, exactly as is observed for the flying oscillation.

Therefore, distinct from the low ${\it\beta}$ case discussed above, the horizontal force perturbations on the oscillating foil are not synchronous with the primary oscillation, and so actually feed back positively on perturbations in the foil’s horizontal location, leading to substantial quasi-periodic movement of the flying oscillating foil. In figure 29 we plot the trajectory of the flying foil, showing just such a non-trivial quasi-periodic horizontal motion. Fundamentally, due to the high frequency of oscillation when ${\it\beta}$ is large, and thus the potential for multiple, close frequencies due to vortex–vortex interaction to exist, freedom to move horizontally actually encourages loss of symmetry, making the symmetric flow around intermediate aspect ratio foils less stable in a very real sense. Conversely, at lower frequencies of oscillation, (i.e. smaller ${\it\beta}$ to the right of the freezing point) the synchronization of oscillation, shed vortices and horizontal forces cause freedom of the foil to move horizontally actually to stabilize strongly synchronous asymmetry, through a locked mechanism of negative feedback.

Figure 29. (a) Time dependence of the $x$ -coordinate of the centre of mass of the elliptical foil with aspect ratio $AR=0.5$ undergoing a flying oscillation with parameters $KC=3.2$ and ${\it\beta}=40$ as shown in figure 27(b). (b) Horizontal trajectory of the mass centre at $t/T_{0}=80{-}100$ . Some specific time instants are marked with various symbols on the two panels.

4.3. Suppression of primary $\text{QP}_{\text{L}}$ -type asymmetry for flying oscillations

As is apparent in figure 20(c,d), allowing a foil of aspect ratio $AR=0.1$ to move horizontally markedly changes the transition boundary in the region of parameter space where the new primary $\text{QP}_{\text{L}}$ -type asymmetry, as discussed in § 3.3, occurs. To investigate how this type of asymmetry is modified by allowing the foil to fly, we consider the flow with parameters $AR=0.1$ , $KC=1.26$ and ${\it\beta}=60$ . As is shown in figure 20(c), the flow with these parameters is symmetric when the foil is purely oscillating, but is asymmetric when the foil is allowed to fly. In figure 30, we plot vorticity contours for these two flows when the foil is at its minimum vertical location. The flying oscillation flow shown in figure 30(b) is undoubtedly asymmetric, but by comparison with figure 11, it bears much more qualitative resemblance to the S-type asymmetry flows shown in figure 11(c,d) than to the primary $\text{QP}_{\text{L}}$ -type asymmetry flows shown in figure 11(a,b).

Figure 30. Twenty-two evenly spaced contours of vorticity between $-500$ (blue) and $+500$ (red) for flow around an elliptical foil with aspect ratio $AR=0.1$ when the foil is at its minimum vertical location for flows with parameters $KC=1.26,{\it\beta}=60$ subject to (a) a pure oscillation and (b) a flying oscillation.

Figure 31. Time dependence of the horizontal force on an oscillating elliptical foil with parameters $AR=0.1$ , $KC=1.26$ and ${\it\beta}=60$ for: (a) a pure oscillation; and (b) a flying oscillation. Time is scaled with the period of the primary oscillation $T_{0}=1/{\it\beta}$ . Note the relative size of the vertical axes.

This qualitative resemblance can be made more quantitative by considering the time dependence of the horizontal force on the foil, as shown in figure 31. The horizontal force on the foil with asymmetric flow exhibits an initial transient, which overshoots somewhat before settling back to its final, still significant quasi-steady state, as shown in figure 31. Although the force undoubtedly oscillates, there is no evidence of ‘beating’ in its amplitude. The power spectrum of the horizontal force time history, as shown in figure 32(b), is completely dominated by twice the frequency of the primary oscillation frequency $2f_{0}=2{\it\beta}$ .

Figure 32. Horizontal force power spectra for the flow with a purely oscillating elliptical foil with parameters $AR=0.1$ , $KC=1.26$ and ${\it\beta}=60$ for: (a) a pure oscillation; and (b) a flying oscillation corresponding to figure 31(a,b) respectively. Frequencies are scaled with the non-dimensional frequency of the primary oscillation $f_{0}={\it\beta}$ .

Therefore, all the evidence points towards the primary $\text{QP}_{\text{L}}$ -type asymmetry of the purely oscillating low aspect ratio foil being suppressed when the foil is allowed to move. Indeed, the transient in the horizontal force in this flow is associated with the foil accelerating to a constant speed of horizontal locomotion. It appears for this aspect ratio that the onset of asymmetry for flying oscillations inevitably leads to locomotion. Forcing the foil to remain at a fixed location leads to the primary $\text{QP}_{\text{L}}$ -type of asymmetry for high values of ${\it\beta}$ , but once the foil is free to move, the growth mechanism for this type of asymmetry is completely removed.

On the other hand, unsurprisingly for the essentially symmetric purely oscillating foil, as shown in figure 31(a), the horizontal force is very small in magnitude, and is clearly highly periodic. That periodicity corresponds closely to the primary frequency of the oscillating foil, as shown by the power spectrum plotted in figure 32(a). We have considered flows associated with other parameter combinations along the transition boundary for flying oscillations of foils with $AR=0.1$ , and, once asymmetry onsets, the asymmetry is always of S-type. We find that the foil always flies at an eventually close to constant speed and we have been unable to identify a parameter combination which leads to primary $\text{QP}_{\text{L}}$ -type asymmetry for flying oscillations.