2. Description of the experimental set-up

The experiments were carried out in the free-surface recirculating water channel of the Thermo-fluid Dynamic Systems and Microsystems Research Group of the Polytechnic University of Madrid (UPM). The water channel width and depth at the test section are 900 and 830 mm, respectively. The inflow conditions at the test section can be controlled in terms of mean speed, uniformity and low turbulence. A rigid prism with a fixed cross-section ($A$), 20 mm in diameter $D$ was attached vertically to the free end of a double-blade elastic system. The submerged length of the prism was $L=210$ mm (see figure 1). The elastic system, which follows the arrangement described by Assi et al. (Reference Assi, Meneghini, Aranha, Bearman and Casaprima2006), was made up of two parallel rigid aluminium blocks ($B$), coupled to a pair of thin spring-steel flexor blades ($C$). This elastic system acts as the prism support and provides a linear restoration response. In this way, the prism is restricted to oscillate in the direction transverse to the flow ($y$) with low mechanical damping. The experimental set-up is similar to that used in previous research by the authors (see, e.g., Xu-Xu, Barrero-Gil & Velazquez Reference Xu-Xu, Barrero-Gil and Velazquez2016; Vicente-Ludlam, Barrero-Gil & Velazquez Reference Vicente-Ludlam, Barrero-Gil and Velazquez2018). During the experiments, the transverse displacement of the prism was measured by a non-contact laser vibrometer ($D$) from Aquity AR500-500 (range 500 mm, resolution 0.05 mm) whose DC signal was registered through a Ni-USB 6002 DAQ card from National Instruments. The sampling frequency was 90 Hz. The total mass of the system was controlled by adding masses to the lower block of the elastic system ($E$). In all experimental tests, the total mass of the system was $m=2.4$ kg, the mechanical damping was $c=0.0842\ {\rm N}\ {\rm s}\ {\rm m}^{-1}$ and the stiffness $k=98\ {\rm N}\ {\rm m}^{-1}$. Mechanical damping and stiffness were measured via free vibration tests in still air. The frequency of oscillation in still air was found to be $f_N=1.01$ Hz. In dimensionless terms, the mass ratio was $m^*=m/\rho D^2 L=28.5$ ($\rho$ is the water density) and damping ratio $\zeta =0.0028$. The flow speed $U$ was varied between $0.05\ {\rm m}\ {\rm s}^{-1}$ and a maximum of $0.3\ {\rm m}\ {\rm s}^{-1}$ (Reynolds number $Re=UD/\nu$, where $\nu$ is the kinematic viscosity of the water, between 1000 and 6000), as summarised in table 1.

Figure 1.

Sketch of the experimental set-up (dimensions are in millimetres). (a) Top and side view of the water channel. (b) Close view photograph of the elastic system. (c) Detail of the elastic system's deformation when a translation in the transverse direction is applied.

Table 1.

List of parameters during experiments.

In some experiments, PIV visualisations were performed to characterise the flow field of the prism wake. The flow was seeded with hollow glass spheres having a mean diameter of $10\ {\mathrm {\mu }}{\rm m}$ (HGS-10). Illumination was provided by a diode-pumped solid-state 532 nm continuous laser ($F$) from CNI Lasers with a maximum power of 5 W, that produced a 3 mm thick horizontal planar laser sheet. Images were taken using high-speed camera global shutter (model TIS-DMK-37BUX252) with a resolution of $2048\ {\rm pixels}\times 1536\ {\rm pixels}$. The camera was equipped with a 25 mm lens, and was positioned below the bottom side of the water channel $(G)$, which is made of glass, with one of the axis of the sensor of the camera aligned with the flow direction. A magnification of approximately $9.9\ {\rm pixels}\ {\rm mm}^{-1}$ was obtained. The field of view of the camera corresponded to a rectangle of $10.5\ {\rm D}\times 7{\rm D}$ that had physical dimensions of $209\ {\rm mm}\times 139\ {\rm mm}$, and focused on the cylinder near wake. PIV image cross-correlation analysis was carried out with open-source PIVlab software (Thielicke & Stamhuis Reference Thielicke and Stamhuis2014). Each PIV area was divided into smaller sub-interrogation areas of $64\ {\rm pixels}\times 64\ {\rm pixels}$. Through a parallel self-consistent re-computation of the flow field in successive interrogation areas of $32 \ {\rm pixels}\times 32\ {\rm pixels}$ and $16\ {\rm pixels}\times 16\ {\rm pixels}$, with 50 % overlap, spatial resolution was refined (final computations were consistent with the flow fields obtained for larger windows). The final spatial resolution of the flow field for the experimental tests was of the order of 2 mm, which is considered to be sufficient to obtain information of the near-wake vortex shedding structures.

Twelve different polylactic acid (PLA) prisms, with the same length, were manufactured on a 3D printer, and tested in the water channel. As already stated, the submerged length of the prisms was 210 mm, so the aspect ratio $L/D$ was 10.5. The geometries of the prism's cross-sections were selected accounting for previous studies performed by other researchers in the area of energy harvesting from VIV/TG. Some variations of those standard geometries were also attempted. In all cases, the transverse dimension was the same (20 mm). All cross-sections had a plane of symmetry aligned to the incoming flow (see figure 2).

Figure 2.

Photograph of the prisms tested. In all cases, the transverse (normal to flow) dimension is 20 mm. From left to right and upper to bottom: circular cylinder, D-section, C-section, diamond, PTC cylinder, rectangle (10 mm short side), square, cylinder with splitter plate (12.5 mm in length), trapezoid (20 mm along the flow direction and 10 mm short side), equilateral triangle, semi-cylinder plus triangle ($90^{\circ }$ main vertex) cir-tri and open equilateral triangle.

All the experiments were carried out as follows. First, the incoming water velocity was prescribed at a low value ($0.05\ {\rm m}\ {\rm s}^{-1}$), and measurements were taken during a time span of 2 min. Then, the instantaneous position of the prism $y(t)$ was measured and registered. After that, the water speed was increased in steps of $0.005\ {\rm m}\ {\rm s}^{-1}$ and the measurement procedure was repeated until either a speed of $0.3\ {\rm m}\ {\rm s}^{-1}$ was achieved, or the prism amplitude was higher than 60 mm (this limit was prescribed to ensure linearity in the elastic response). The following variables were computed for each water velocity: (i) the normalised amplitude of oscillation $A^*=A_{10}/D$, where $A_{10}$ is the average of the of the top 10 % of oscillation amplitude peaks; (ii) the normalised frequency of oscillations $f^*=f/f_N$, where the frequency of oscillations $f$ was computed using the fast Fourier transform of $y(t)$; and (iii) the efficiency of energy transfer from the flow to the body. The latter was computed by dividing the mean power dissipated by damping by the kinetic power of the incident flow across the submerged prism area:

(2.1)\begin{equation} \eta=\frac { \dfrac{1}{T}\displaystyle\int_0^T c\dot{y}^2\,{\rm d}t} {\dfrac{1}{2}\rho U^3D L}, \end{equation}

where $T$ is a time period for averaging and $c$ is the mechanical damping constant. The transverse velocity of oscillations $\dot {y}$ was obtained by numerical differentiation of the transverse position $y$.

4. Maximum efficiency as a function of the geometric center of the cross-section and afterbody length

It is well known that transverse FIV depends both on the size and on the shape of the afterbody (Parkinson Reference Parkinson1989). The afterbody is defined as the part of the cross-section located downstream of points where flow detachment is expected to occur. As only geometries with a symmetry axis along the flow direction are prone to experience strong VIV and TG, the centroid position of the semi-cross-section, having coordinates $x_c,y_c$ with respect to the stagnation point $S$, see figure 11, is selected here to characterise the shape. The length of the afterbody is given by $x_s$ and the total length by $x_t$. Thus, two dimensionless parameters are introduced, namely $\tau _1=x_c/(x_t+x_{s})$ and $\tau _2=y_c/(x_t+x_s)$. Note that $\tau _1$ provides a measure of the frontsize length versus the afterbody length. As $\tau _1$ increases, the frontbody length dominates over the afterbody length. Here $\tau _2$ is a measure of the bluntness of the cross-section shape. Small values of $\tau _2$ are expected for streamlined shapes and higher values for blunt shapes. Figure 11 shows a contour plot computed with the measured peak efficiencies for all geometries tested as a function of $\tau _1$ and $\tau _2$.

Figure 11.

Contour plot of the peak efficiency as a function of $\tau _1$ and $\tau _2$.

In spite of the limited number of geometries tested, it seems from visual inspection that there appears two different zones with good peak efficiencies $\eta _M$ around $\tau _1=\tau _2=0.22$ and around $\tau _1=\tau _2=0.28$. The first region corresponds to solid cross-section shapes (e.g. D-section) whereas the second area corresponds to open cross-section shapes (such as C-section).). For the first region, it could be noted that a quasi-linear relationship seems to be present between $\tau _1$ and $\tau _2$ for optimal $\eta _M$. Note that there is a line (see line ${MM}'$ in figure 11) that shows a monotonic trend, from lower to higher $\eta _M$ in the M to ${M}'$ direction. However, we have not been able to find the same type of trend for the second region, so we suggest to consider the line ${MM}'$ as a locus for high efficiency (let us consider it as ‘optimal’). A linear fit of the optimal line ${MM}'$ gives

(4.1)\begin{equation} \tau_2^{op} \approx 1/2-(5/4)\tau_1^{op},\end{equation}

where ‘op’ superscript denotes ‘optimal’. From the definitions of $\tau _1$ and $\tau _2$ it follows that

(4.2a,b)\begin{equation} \tau_1=\frac{x_c}{x_t+x_s}=\frac{x_c}{x_t(1+x_s/x_t)}=\frac{x_c}{x_t}\frac{1}{1+\beta}, \quad \tau_2=\frac{y_c}{x_t}\frac{1}{1+\beta}, \end{equation}

where $\beta =x_s/x_t$ is a dimensionless parameter that relates the length of the afterbody to the total length of the cross-section shape in the flow direction. Substituting (4.2a,b) into (4.1) yields

(4.3)\begin{equation} \frac{5}{4}\frac{x_c^{op}}{x_t}+\frac{y_c^{op}}{x_t}=\frac{1+\beta}{2},\end{equation}

that gives the location of the centroid for optimal energy transfer (that is along the line ${MM}'$ in figure 11) for a fixed $\beta$ value (afterbody length versus total length).

To test the validity of this relationship, the optimal cross-section for the half-elliptical and isosceles triangular families was sought. Considering, first, a half-ellipse of transverse dimension $D$ and an in-line dimension $a$ (see figure 12 for illustration), the position of the centroid in the semi-cross-section is given as $x_c=4a/(3{\rm \pi} )$ and $y_c=2D/(3{\rm \pi} )$, $x_t=a$ and $x_s=a$ ($\beta =1$). From (4.3) it follows that for optimal energy transfer $5+2D/a=3{\rm \pi}$ and, therefore, $a=2D/(3{\rm \pi} -5)$. For $D=20$ mm the in-line dimension $a$ has to be of 9.04 mm. Figure 12 shows the normalised amplitude of oscillations and the energy transfer efficiency as a function of the reduced velocity for the half-ellipse with $D=20$ mm and $a=9.04$ mm. A peak efficiency of 0.47 was found, which is larger that the value found for the D-section (which is a half-ellipse with $D=20$ mm and $a=10$ mm). A half-ellipse with a lower aspect ratio $D=20$ mm and $a=8$ mm was also manufactured and tested in order to see whether maximum efficiency is lower than that for the optimal case. As can be seen in figure 12, the maximum efficiency is in this case 0.42, a lower figure than that obtained for the ellipse with $D=20$ mm, and $a=9$ mm (it was 0.47). This result is thus inline with the prediction that for half-ellipses the optimum should be achieved when the aspect ratio is close to 0.45 ($D=20\ {\rm mm},\ a=9.04\ {\rm mm}$).

Figure 12.

Normalised amplitude of oscillations (a) and efficiency $\eta$ (b) dependence with the reduced velocity for the half-ellipses with in-line dimensions $a=9$ mm and $a=8$ mm.

For isosceles triangular cross-sections of transverse dimension $D$ and in-line dimension a (see figure 13), $x_c=a/3$ and $y_c=D/6$, $x_t=a$ and $x_s=a$ ($\beta =1$). Then, from (4.3) it follows $5/12+D/(6a)=3{\rm \pi}$ and, therefore, the optimal triangle would be that with an in-line dimension of $a=2/7 D$. This implies a main vertex of approximately $120^\circ$. Figure 13 shows the normalised amplitude of oscillations and the efficiency as a function of the reduced velocity for the isosceles triangle with main vertex of $120^\circ$. A very high value of efficiency is found with a maximum figure around 1.15 at reduced velocity 8.1 (this maximum value involves some ambiguity because oscillations were so strong that it was decided not to increase flow speed further to avoid nonlinearities in the elastic response). This high efficiency value figure agrees well with the theoretical prediction about optimal shape. In order to test if best isosceles triangle was found, another isosceles triangular cross-section triangle with a main vertex of $90^\circ$ was manufactured and tested. Results are given in figure 13. In terms of efficiency a maximum value $\eta _M$ of 0.39 was measured which, as expected, is lower than that measured for the $120^\circ$ isosceles triangle. These results suggest that taking the position of centroid may be useful to characterise the expected flow energy transfer from the flow to the body, and that the information given by the $\tau _1$–$\tau _2$–$\eta _M$ map describes a correct trend when looking for the optimal cross-section shape.

Figure 13.

Normalised amplitude of oscillations (a) and efficiency $\eta$ (b) dependence with the reduced velocity for the half-ellipse and isosceles triangles ($120^\circ$ and $90^\circ$ main vertex).

Additional experiments were carried out to further test the validity of the coordinates $\tau _1$ and $\tau _2$ to link with $\eta _M$. A polygonal cross-section, (see figure 14 for geometric definition) was designed and manufactured for testing purposes. For this geometry $tau_1=0.211$ and $\tau _2=0.214$, that are very close to the ones of the D-section ($\tau _1=\tau _2=0.212$). Figure 14 shows the results obtained in terms of efficiency. In spite of the geometrical differences between the D-section and the new polygonal cross-section (note the presence of a corner in the afterbody of the polygonal cross-section), $\eta _M$ is quite similar in both cases ($\eta _M=0.45$ for the new triangle and $\eta _M=0.43$ for the D-section). This reinforces the idea that $tau_1$ and $\tau _2$ are useful coordinates to characterise $\eta _M$.

Figure 14.

Power efficiency as a function of the reduced velocity for the D-section and a polygonal cross-section. Modified triangular cross-section presents nearly the same values of $\tau _1$ and $\tau _2$ than the D-section.

5. Concluding remarks

In a simplified way, a system meant to extract energy from a fluid flow that takes advantage of transverse FIV is made up of three main elements: (i) the geometry of the body immersed in the fluid flow; (ii) the elastic (structure) system that provides the restoring force; and (iii) the transducer, where the oscillatory mechanical power is transformed in electric power. Searching for energy transfer optimisation from the ‘fluid side’, thus avoiding structural and energy optimisation stages, requires to find the body with the optimal cross-section shape. In this context, hydroelastic experiments have been carried out to study the influence of the cross-section geometry in the energy extraction from transverse FIV. For all tested geometries all relevant dimensionless parameters: the mass ratio, the dimensionless damping, the reduced velocity and the Reynolds number remained the same. Because of the low value of the mass damping parameter ($m^* \zeta =0.079$) all geometries were unstable to transverse FIV in the range of reduced velocities actually covered. By doing this, the role of the cross-section geometry can be isolated, and its influence inferred correctly.

In a first set of experiments, twelve cross-section shapes, commonly considered in the past in the study of transverse FIVs, were tested. It was found that, for this set of geometries, the most unstable shape that extracts energy from the flow more efficiently is the D-section shape (half-cylinder), with a maximum efficiency $\eta _M$ of 0.44. As a general trend, excluding the circular cross-section, it was found the more prone to TG the shape the higher the efficiency achievable.

To better understand the role of the shape on $\eta _M$ two dimensionless geometric parameters ($\tau _1,\tau _2$) were introduced to characterise the cross-section shape. These parameters were linked to the centroid position of the semi-cross-section, and to the length of the afterbody, and they map $\eta _M$ vs $\tau _1$ and $\tau _2$. In spite of the relatively limited number of shapes tested in the study, this bi-parametric map allows the identification of regions where $\eta _M$ is high. A further step was then given noting that there seems to exist a locus in the $\tau _1$ and $\tau _2$ space where maximum efficiency is optimised, which is close to a linear mathematical relationship between $\tau _1$ and $\tau _2$. This optimal relationship between $\tau _1$ and $\tau _2$ allowed to propose ‘in advance’ the best cross-section in the family of isosceles triangles, which resulted to be that with a main vertex of $120^\circ$. Experiments confirmed the expected high values for energy transfer for this case, where a very high figure of maximum efficiency was measured (1.15), and supports the validity of the proposed relationship. Also they supported the idea that the geometric centroid of the semi-cross-section and the afterbody length of the cross-section could be taken as a simple and effective metric to characterise the energy transfer and the magnitude of transverse oscillations. It should be noted that the $\tau _1$–$\tau _2$–$\eta _M$ map is useful not only to look for a promising cross-section shape for energy-harvesting purposes, but, also, to search for ‘safe’ cross-section shapes against FIV (those giving low values of $\eta _M$). For instance, the map could be useful to identify safe FIV regions, which can be useful in the offshore and wind engineering fields.

Finally, it should be observed that the $\tau _1$–$\tau _2{-}\eta _M$ relationship has been obtained by isolation of the geometry effects, because the other governing parameters of the problem were unchanged during experiments (mass ratio, dimensionless damping, reduced velocity range, aspect ratio and Reynolds number range). Nevertheless, a word of caution is required here because the question as to whether the $\tau_1{-}\tau_2{-}\eta_M$ map trend remains quantitatively valid for other combination of mechanical and flow parameters is still open, and requires further research.