Improving the performance of galloping micro-power generators by passively manipulating the trailing edge

Recent trends in distributed sensing networks have generated significant interest in the design of inexpensive and scalable local power generation units. One such energy harvesting method exploits aerodynamic instabilities such as galloping to harness energy from a moving fluid. In particular, as shown in Fig. 1(a), when a fluid moves past a harmonic oscillator, the flow separates at the leading edge of the prism causing inner circulation bubbles to form under the resulting shear layers. For certain prism shapes, the circulation can produce negative surface pressure which causes a net lift on the prism. The net lift breaks the symmetry between the shear layers on the top and bottom surfaces which produces more lift. The process continues until the energy fed to the structure by the fluid balances all energy dissipation mechanisms. This results in a steady-state fixed-amplitude periodic motion of the prism known as limit-cycle oscillations. The resulting oscillations can be utilized to strain a piezoelectric element and/or move a magnet with respect to a fixed coil, which, in turn, channels a periodic current, I(t), to an electric load R_l, thereby providing a scalable and effective energy transduction mechanism.

The performance of a galloping flow energy harvester (GFEH) is assessed by two criteria: (i) The cut-in wind speed (speed at which the limit-cycle oscillations are initiated) which should be minimized; and (ii) the output power as a function of the wind speed which should be maximized.^1–6 In this letter, we propose to improve upon these two metrics by manipulating the flow at the trailing edge of the prism. Flow control at the trailing edge has been widely utilized previously to decrease drag, reduce noise, enhance mixing, and suppress vibrations. Here, we show that attaching a rigid splitter plate, or a tail fin, to the prism can create flow patterns around the harvester which are favorable for energy generation. This idea is motivated by the work of Assi and Bearman⁷ who demonstrated that the placement of a streamwise-oriented plate behind a circular cylinder amplified its galloping response.

The influence of the splitter plate on the galloping response can be understood by considering Fig. 1(b) in conjunction with Bearmans work in which he examined the flow behind a stationary blunt cylinder with splitter plates of varying lengths.⁸ Bearman observed that, as the flow separates at the leading edge, it creates an unstable shear layer which grows in size as the flow progresses by entraining irrotational fluid from the surrounding region. If the prism has a long after body, the shear layer attaches to the surface of the body at a point where the flow has a zero velocity gradient normal to the surface.⁹ Bearman showed that the addition of the splitter plate increases the shear layer curvature and can trap a large recirculation bubble between the backward face of the body, the splitter plate, and the impinging shear layer. This recirculation bubble can improve lift and hence enhance the galloping response of the system. It was also observed that the lift force is maximized at an angle of attack known as the reattachment angle where the shear layer first reattaches to the side face of the body.^10,11 As the body is rotated past the angle of reattachment, α_re, the recirculation bubble trapped underneath the shear layer shrinks, dramatically reducing the lift force.

To understand the relationship between the reattachment angle and the lift force, we use the quasi-steady approximation of the aerodynamics force to approximate the lift force on the prism. The quasi-steady assumption states that a galloping body can be modeled as experiencing the same aerodynamic forces as those acting on a static body positioned at an equivalent angle of attack. The accuracy and validity of this assumption require that the movement of the body is sufficiently slow with respect to how quickly the wake is swept downstream.¹² Under the quasi-steady assumption, the transverse aerodynamic force, f_a, acting on the body is given by

where ρ and U are the density and free velocity of the flow, respectively, A is the frontal area of the bluff body at zero incidence, and C_a is the transverse aerodynamic coefficient that is a function of the angle of attack

The angle of attack is used to describe the angle at which the flow impacts the body for non-zero body velocity $ẋ$⁠.

Since the lift force is directly proportional to $Ca(α)$⁠, the magnitude of galloping oscillations depends on the shape of the C_a curve. Figure 2 illustrates an example of such curve for a square prism. The aerodynamic forces continue to “pump” up the amplitude of the harvester's response only as long as sign of the oscillator's velocity $ẋ$ is the same as that of C_a which occurs in quadrants I and III. When the amplitude of oscillation grows very large and the body's velocity $ẋ$ is pushed outside of the range in which it agrees in sign with C_a, the aerodynamic force begins to act against the direction of motion, forcing an upper limit on the response amplitude.

The peak in the C_a curve corresponds to the reattachment angle, α_re, beyond which the lift decreases sharply and the C_a curve changes sign. Therefore, to increase the galloping response, it is essential to increase the reattachment angle as much as possible and, when possible, reduce the rate at which the C_a curve drops beyond the reattachment angle.

The most straightforward way to increase the reattachment angle is to taper the afterbody such that the side faces of the prism are pulled away from the shear layer. As a result, a larger rotation of the prism becomes necessary before the side faces are in close enough proximity for the shear layers to reattach.¹⁰ Nonetheless, the large distance between the side faces and the shear layers resulting from tapering the afterbody reduces the interaction between the body and the flow substantially at small angles of attack. Therefore, initiating oscillation at small angles of attack becomes very difficult which significantly increases the cut-in wind speed of the harvester.

An ideal galloping oscillator for energy harvesting must therefore combine the at-rest instability of an untapered prism with the high-amplitude response of a more tapered geometry. This goal requires side faces both near the flow to promote at-rest instability, yet far from the flow to allow for reattachment to occur only at high angles of attack. One feature comes at the cost of the other. In this letter, we show that this can be realized by introducing a secondary side surface by extending a splitter plate from the trailing face of the bluff profile, as previously shown in Fig. 1(b).

Several experimental trials were run to explore the effect of the splitter plate on the output power. The three profiles shown in Fig. 3 were examined: a square, a trapezoid, and an equilateral triangle. All had a characteristic cross-stream width D = 5 cm and a 10 cm height. Each profile was tested with no plate, a 0.5D plate, and a 0.8D plate. The plate was fixed to the base of the bluff body at the point of attachment to the beam. The natural frequencies of all systems were designed to be the same.

Each prism was mounted on a mild steel beam of 21.5 cm length, 3 cm width, and 0.635 mm thickness. A Smart Materials M8528 P2 Macro Fiber Composite strip was glued to the upper surface of the beam to serve as a voltage generator, and a 593 kΩ resistance was connected in parallel with the MFC patch to serve as the electric load. The response of the harvester was studied in an Aerolab Educational Wind Tunnel (EWT), having a 30.5 cm square test chamber cross section. The blockage ratio was 5%. Oscillation amplitude of the body was limited to avoid interactions with the boundary layer on the tunnel walls.

The results are presented in Fig. 4. It is evident that, in most cases, the output power improves substantially upon the addition of a splitter plate. The square prism experienced a maximum improvement of 27% in the output power when a 0.5D splitter plate was added. The maximum power improvement increased further to 67% with the 0.8D splitter plate. The trapezoidal prism experienced 28% maximum improvement in the output power with the 0.5D plate and 24% improvement with the 0.8D splitter plates. An optimal splitter length therefore exists for the trapezoidal prism because the 0.8D splitter plate brings the performance down from the 0.5D case.

Adding a splitter plate also affected the cut-in wind speed. The triangular prism without the plate and the one with the 0.5D plate did not gallop from rest; i.e., without subjecting the prism to a large initial displacement. However, with the 0.8D tail, the body was able to gallop from rest. Furthermore, the trapezoidal prism without a splitter plate did not exhibit steady-state galloping behavior until 4.5 m/s, while the trapezoids with the plates galloped from rest at much lower speeds.

The presence of an optimal plate length for the trapezoidal prism requires further investigation. To this end, the flow patterns around the harvester with the trapezoidal prism were simulated. Transient fluid flow simulations were carried out by using ANSYS Fluent. The transition Shear Stress Transport (SST) model was employed for the transient flow with a Reynolds Number (Re) of 22 000 and a time step of 0.001 (for consistency with Parkinson and Smith¹³).

The transient computational analysis shows that, after the fluid flow is fully developed, the flow around the structure undergoes a stable periodic transition between its maximum and minimum lift instances. Using the ANSYS simulations, the traverse lift coefficients C_a with and without splitter plates were obtained and compared as depicted in Fig. 5. The C_a curves shown in Fig. 5 correspond to the average lift force within a period of oscillation of the fluid flow.

Simulations reveal that the reattachment angle corresponding to the peak of the C_a curve increases from around $18°$ in the absence of the splitter plate to somewhere around $21°$ when a splitter plate of 0.5D is added. As described previously, an increase in the reattachment angle improves the performance of the harvester. Thus, the numerical results agree with the experimental findings which clearly illustrate that the trapezoidal prism with the 0.5D plate outperforms the one without the splitter plate.

When the length of the splitter plate is increased to 0.8D, the reattachment angle drops back to about $19°$⁠. Hence, as shown experimentally, the output power of the harvester drops when compared to the 0.5D case. Nevertheless, even though the reattachment angle drops to almost the same value as in the no-splitter case, the harvester with the 0.8D splitter plate still outperforms the harvester without the plate because, as shown in Fig. 5, the lift coefficient, C_a, is much larger in the 0.8D case.

The lift coefficient shown in Fig. 5 is governed by the flow pattern whose main characteristics can be seen in Fig. 6. When approaching the leading edge of the prism, the flow separates creating two shear layers that form on either side of the prism as shown in Fig. 6. Naturally, upon separation, these two layers tend to curve towards one another in order to recreate a uniform steady stream. The curvature of the shear layers is governed by the angle of attack and the Reynolds number of the flow.¹⁴ The shear layer forming on the upper side of the prism (rotated away from the flow) has less curvature and hence can trap larger circulations under it. On the other hand, the shear layer forming on the lower side (rotated towards the flow) has a larger curvature and therefore traps smaller circulations. The relative size, location, and flow velocity within these circulations determine the pressure distribution on the surface of the body. Maximum lift occurs when the pressure difference between the upper and lower faces is maximized.

As shown in Fig. 6, the upper shear layer traps a large circulation, CB4, under it which creates upward suction. However, this circulation is far away from the prism and hence has little influence on the pressure on the top surface of the body. Furthermore, a smaller circulation bubble, CB3, forms right on the upper face of the prism. This bubble forms when the reverse flow resulting from CB4 separates at the upper rare corner of the prism. The size of CB3 is determined by velocity of the streamlines approaching the upper rare corner with larger velocities resulting in a larger CB3. The velocity, in turn, depends on the degree of interaction between the two curving shear layers near the end of the splitter plate. The more the upper streamlines interact with the lower shear layer, the slower the velocity of the streamlines. Since for longer splitter plates the interaction is reduced, CB3 is generally larger.

As the lower shear layer curves towards the body, it traps two other circulation bubbles above it. For the case shown in Fig. 6, the shear layer traps two high-velocity circulation bubbles which create very large downward suction on the body. The first one, CB1, is trapped between the shear layer and the lower face of the prism while the second, CB2, is trapped between the splitter plate, the shear layer, and the lower back face of the body. The presence of CB2 produces a large downward suction that aids in lift production. Note that in the absence of the splitter plate, CB2 does not form which reduces the net lift on the prism.

Based on this understanding, we can now relate the velocity streamlines around the trapezoidal prism to the lift production for the 0.8D and 0.5D splitter plates. In both cases, the streamlines were generated for angles of attack around the peak of the C_a curve. Since the fluid patterns around the harvester repeat periodically, the velocity streamlines vary periodically too; that is, the characteristics of the velocity streamlines obtained at the minimum lift are similar to those observed at the maximum lift. As such, for clarity and brevity, only velocity streamlines associated with the minimum lift are shown here. Figure 7(a) depicts the velocity streamlines for the 0.8D case at a $19°$ angle of attack. In this case, the shear layer attaches to the lower surface of the body at the tip of the splitter plate. Although there is a slight curvature of the shear layer near the lower rear corner of the trapezoidal prism, the shear layer does not attach to the prism itself. As such, a large circulation bubble, CB1, forms below the shear layer creating a very low pressure zone. Furthermore, since at $19°$⁠, CB3 does not form and CB4 is far away from the splitter plate, very little upward suction occurs on the top face. This produces perfect (downward) lift conditions: a large circulation bubble on one surface and no circulation on the opposing surface. Therefore, as shown in Fig. 5, the trapezoidal profile with the 0.8D splitter plate produces more lift at this angle of attack than any other angle.

When the angle of attack is increased to $21°$⁠, the lower surface rotates further towards the flow causing the lower shear layer to attach at two points, one near the lower rear corner of the prism and the other near the tip of the splitter plate. The reattachment results in the formation of two smaller circulation bubbles, CB1, and CB2. These circulation bubbles are countered by a circulation bubble, CB3, which forms on the upper surface. In addition, CB4 moves closer to the upper surface after reattachment at the two points. The result is that the lift force drops sharply when compared to the $19°$ angle of attack. A further increase in the angle of attack to $23°$ increases the size of the circulation bubble, CB3, even further which reduces the lift coefficient to negative values.

When comparing the results of the 0.5D splitter plate, Fig. 8, to those obtained using the 0.8D splitter plate, we note that the shear layer does not attach to the lower surface of the 0.5D splitter plate at the $19°$ angle of attack. As described previously, when the flow separates at the leading edge, the shear layer forms, increases in size, and then curves back in order for the flow on either side of the bluff body to reattach into one uniform stream. However, the presence of a splitter plate prevents the shear layers from rejoining. As a result, the shear layer attaches to the splitter plate instead. A longer splitter plate naturally allows the shear layer to attach at smaller rotation angles. Hence, for the longer splitter plate, the flow reattaches at 19° whereas it needs a larger rotation angle to reattach in the case of the 0.5D splitter plate. Despite the inability of the lower shear layer to reattach to the tip of the splitter plate, a circulation bubble, CB2, remains trapped between the shear layer, the splitter plate, and the rear surface of the trapezoid which aids in lift production.

As the angle of attack is increased to $21°$⁠, Fig. 8(b), the shear layer reattaches to the corner of the prism creating a small circulation bubble, CB1. This, combined with CB2, serves to improve lift production. Unlike the streamlines associated with the 0.8D splitter plate, CB3 does not form on the upper surface because the lower shear layer has more interaction with the upper shear layer for the shorter splitter plate which slows down the reverse flow on the top surface. The absence of CB3 improves the lift substantially, Fig. (5). A further increase in the angle of attack to $23°$ results in the formation of a small circulation CB3 bubble at the upper surface which slightly reduces the net lift as shown in Fig. 5.

In summary, this letter demonstrates that attaching a rigid splitter plate to the rear face of a galloping prism extends the useful range of the aerodynamic instability for larger oscillation amplitudes and power output. Experimental results demonstrate as much as 67% power enhancement for the square prism and a significant reduction in the cut-in wind speed of the generator for all the investigated geometries.