Chapter III: Characterization of the Behaviour of an Airfoil Driven in the
3.10 Summary of Airfoil Interactions with Oncoming Vorticity
away from the centerline that created jet-like flow strong enough to overcome a region of net drag directly behind the airfoil, which leads to a positive mean momentum flux. It is interesting to note the emergence of an opposite pattern here, linked to the upstream orientation of the generated pairs of counter-rotating vortices, most clearly seen in Figure 3.15 in the following section. Computing mean momentum flux at each station in Figure 3.14, we also find that when the airfoil is present, the mean momentum flux is slightly reduced compared to that for the cylinder-only case, although this calculation is highly sensitive to the mean flow velocity on a particular day of testing. This is consistent with the idea that the foil extracts net energy from the flow, or that not all of the available momentum harvested from oncoming vorticity is used to generate net thrust.
Figure 3.15: Summary of interactions between the airfoil and vorticity shed by the upstream circular cylinder over one vortex shedding cycle. Top three panels show idealized airfoil position 𝑦 (top left), 𝐶𝐿, 𝑦¤ (top middle), and 𝐶𝑃, 𝐶𝑇 (top right).
Trends in these quantities are indicated in the concentric circles in the bottom panel, between indicated phase locations 1-4. Images in the bottom panel show a simplification of both cylinder and airfoil-derived vorticity and their approximate interactions at each phase location in one vortex shedding cycle. Vortices are labelled according to their origin and sign of rotation. CV: Cylinder Vortex; TEV: Trailing- Edge Vortex; BLV: Boundary-Layer or other Near-Airfoil Vortex (including LEVs, and any vorticity due to flow separation over the airfoil’s surface). Negative (CW) vorticity; Positive (CCW) vorticity. In addition, cylinder vortices have arrows indicating sign of vorticity.
phase relative to oncoming shedding than that of the observed lift and commanded velocity. This shift reflects an offset between the phase angle giving rise to the highest lift magnitude (approximately 𝜙/2𝜋 = 0.25,0.75), and the phase where the airfoil is centered in a region of upwards or downwards flowing fluid. Though noticeable for example in Figure 3.11 showing the𝑦-direction flow velocity (𝑉), the shift is small and it is neglected in Figure 3.15.
The bottom portion of Figure 3.15 shows a simplified picture of the formation mechanism for the 2P wake generated by the airfoil, discussed in Section 3.8 and 3.9. We first consider phase instant 1 in Figure 3.15, when the airfoil is at its minimum position in the frame. A large CW cylinder vortex is located directly over top of the foil, which is is causing a TEV of the same sign to be shed into the flow downstream. In addition, there is a region of separated flow on the foil’s bottom surface (labelled BLV in the Figure). In the wake, trailing edge vorticity is pairing up with a CCW cylinder vortex located downstream and below the foil.
At instant 2 , the oncoming vorticity has advanced through a quarter cycle, such that the large cylinder vortex over top the airfoil in the previous instant is now located just past the trailing edge. The advancement of the upstream CCW cylinder vortex towards the leading edge has energized the flow on the airfoil’s bottom surface, causing previously shed BLV to detach from the leading edge and convect along the foil’s surface towards the trailing edge, initiating flow reattachment. This shed BLV begins to coalesce into a TEV to be shed later in the cycle. The airfoil is moving upwards with its maximum velocity, and a region of flow separation has started to form on the top side near the leading edge. Any separation in the trailing edge region is difficult to distinguish from vorticity due to the large CW cylinder vortex downstream. In the wake, the TEV shed at the previous instant has detached from the airfoil, paired up with a CCW cylinder vortex, and the pair are now convecting downstream and away from the centerline, forming the 2P-type wake.
At instant 3 , a moment representing the dual of instant 1 but where airfoil motion, forces and vorticity are reversed, flow separation over the top surface of the airfoil which was initiated at the previous instant has deepened, with airfoil-derived vorticity remaining localized near the top surface. Cylinder vorticity has advanced such that the airfoil is directly over top of a CCW cylinder vortex and at its maximum position extent. The presence of the CCW cylinder vortex has caused BLV shed at the previous instant to coalesce, and has further fed the formation of a CCW TEV.
This TEV is in the process of pairing with the downstream CW cylinder vortex to
form the next 2P-type wake structure.
Finally, at instant 4 , the dual of instant 2 , we see that the advancement of the oncoming CW cylinder vortex above the airfoil has caused BLV from the top surface of the airfoil to be shed towards the trailing edge as the flow reattaches to the top side. This BLV begins to coalesce into the CW TEV seen in the next instant 1 . The previously shed CCW TEV has paired with the downstream CW cylinder vortex to form a 2P wake structure, which convects downstream and away from the centerline.
This is the second pair of vortices shed per cycle, constituting a 2P wake.
Based on the relationship between the airfoil’s position and the passage of oncoming cylinder vorticity, we see that the airfoil moves to avoid oncoming vortex cores, achieving position maxima as cores are passing through the airfoil’s 𝑥-location.
This appears to correspond to aSlaloming Modeof interaction as identified by Beal et al. (2006). In agreement with the current study, the slaloming mode of interaction was found to give rise to pairs of counter-rotating vortices in the combined wake region, as well as high observed propulsive efficiency (Beal et al., 2006). Though conceptually the propulsive efficiency of our current system is very high, as it extracts thrust without requiring energy input at all and in fact extracts net power, under the conventional definition it would be less than zero (𝜂 < 0). Therefore, this is not a particularly appropriate performance metric for hybrid propulsive/energy extracting systems.
The principal difference between similar past studies and the current experiments, and the key factor in distorting efficiency as a metric for this system, is twofold.
Firstly, in the present case the magnitude of the airfoil’s velocity is much smaller than the maximum magnitude of flow velocity in the region surrounding it. Secondly, the flow and the foil’s velocities are always aligned. These factors together lead to the net power extraction from the flow, since the flow in the region of the airfoil puts energy into the airfoil’s motion, similar to the action of pushing a child on a swing set. This leads to a qualitatively different interpretation of efficiency as a metric, since in the classical framework used to study propulsion, the computed efficiency for our system is always negative. Moreover, for systems of this type, the pole in the equation for efficiency when the required input power to generate thrust in a system passes through zero distorts the interpretation further. This was discussed more completely in Chapter 1.
Despite these challenges in interpretation of similar studies with larger foil ampli- tudes, it is interesting to compare the present vortex formation mechanism with
those discussed in the pioneering work of Gopalkrishnan et al. (1994). In doing so, we see that the mode of interaction considered in the current study represents a different mechanism than those presented previously. Though those authors discuss an ‘Expanding Wake’ mode which has qualitative similarities to the observations made in this study (for example, it also leads to a combined wake of type 2P that convects away from the centerline), in their study vortex repositioning by the air- foil is a dominant mechanism, dragging cylinder vortices across the centerline to pair with previously shed airfoil-derived vorticity. In our study, it is the trailing edge vorticity which crosses the airfoil centerline to pair with downstream cylinder- derived vorticity. The extensive vortex repositioning is made possible by the much larger foil (𝑐 =2𝐷) used in that study, as well as the much larger transverse motion (𝐴0 =0.5𝐷−0.833𝐷). In the present case, the change in pressure field induced by the airfoil is not large enough to affect the trajectories of cylinder-derived vorticity in such an extreme way, and the extent of the wake expansion is correspondingly limited.
In addition, the Expanding Wake mode identified by Gopalkrishnan et al. (1994) appears to correspond most closely to theInterception Modeidentified by Streitlien et al. (1996), though significant differences in assumptions and experimental and/or parametric frameworks between the the studies make a true apples-to-apples com- parison challenging. In the work of Streitlien et al. (1996), the interception mode corresponds to the case where the airfoil encounters vortex cores head on, instead of weaving between them as in the avoidance/slaloming mode. This mode represents a simultaneous maximum in required input power to sustain motion, but results in good efficiency since the thrust produced is also large. In addition, this mode results in the expansion of the combined wake signature by the action of the airfoil.
Although Gopalkrishnan et al. (1994) report variable efficiencies for experiments associated with the Expanding Wake mode (which they found challenging to re- produce, in contrast to the current study), the phase of vortex interaction appears similar, and in both cases an expanding wake signature is the principal feature.
If this interpretation is correct, the present study represents a region of phase space not visualized by Gopalkrishnan et al. (1994). The present experiments, corresponding to the slaloming mode of interaction, operate at a phase of interaction 180° out of phase with the Expanding Wake mode discussed in that study, and hold similarity with the low-power/avoidance mode identified by Streitlien et al. (1996).