Surface protrusion based mechanisms of augmenting energy extraction from vibrating cylinders at Reynolds number 3 × 10³–3 × 10⁴

Marine hydrokinetic energy (MHKE) devices are a class of low head energy conversion systems, which convert the kinetic energy of flowing water in rivers, tidal basins, and ocean waves into mechanical work; which is then converted to electrical power by a suitable power-take off mechanism. They are portable and can be deployed at small start-up costs compared to traditional hydropower.¹ A recent study conducted by the Electric Power Research Institute (EPRI), estimated the recoverable marine hydrokinetic energy from rivers, tidal energy sites, and wave energy resources to be ∼1540 TWh/yr.^2,3 Considering the fact that the total electricity consumed in U.S. in 2011 was 3856 TWh/yr, MHKE systems promise the potential to meet ∼40% of the national electricity demand. MHKE devices can be classified into rotary systems (turbines) and non-rotary (oscillating) systems that behave like point or line absorbers. One such harvesting device, which falls in the non-rotary class and exploits the phenomena of Fluid Induced Motion (FIM) of bluff bodies (especially Vortex Induced Vibration (VIV) and galloping), is called VIVACE (acronym for VIV for Aquatic Clean Energy) and was developed and patented by Professor Michael Bernitsas and co-workers at the University of Michigan.^4–10 The device converts the energy of river/tidal/ocean currents to mechanical energy by inducing free vibration of cylinders^6,11 and can be used in water bodies with current velocities as low as 1–2 m/s, as opposed to turbines which operate in currents with velocities > 2 m/s. A VIV based device is simple in construction and is modular, scalable, and robust and does not employ any rotating parts like turbines or propellers.⁶ 

Historically, VIV has been considered to be detrimental for aero, civil, mechanical, marine, offshore, and nuclear engineering structures; a well-documented example of the destructive nature of VIV is the collapse of the Tacoma Narrows Bridge in Washington in 1940. Consequently, numerous attempts have been made to develop means that would mitigate vibrations due to vortex shedding. A comprehensive review of the various suppression methods tested was given by Zdravkovich.¹² The features added to a cylinder to modify the flow were classified into three different categories, depending on the phenomenological mechanism that was employed. Surface protrusions were used to modify flow separation and the separated shear layers; shrouds were used to tweak the entrainment layer; and near wake stabilizers were used to prevent the interaction of the entrainment layers. Table I presents the added features, successful in suppressing vibration, the phenomenological mechanism that they employ and their directional effectiveness. Figure 1 presents schematics of cylinders with the features discussed in Table I. In the process of investigating suppression mechanisms, some surface protrusion based features; or more appropriately, certain arrangements of some such features proved to be ineffective, as they ended up augmenting the oscillations. These are highlighted in Figure 2. Nakagawa et al.¹³ carried out tests at super critical Reynolds numbers (1 × 10⁶< Re < 6 × 10⁶), with eight helical wires wound around a flexibly mounted cylinder and observed amplitudes twice as high as those observed with the plain cylinder. Their experiments by replacing the eight helical wires with just one¹⁴ also resulted in significant vibrations. Mahrenholtz and Bardowicks¹⁵ achieved enhanced oscillations in the subcritical regime by fitting five fins to a cylinder. They observed maximum excitation for a symmetrical configuration. Gartshore et al.¹⁶ performed experiments in both smooth and turbulent flows by attaching a pair of fins sidewise to a cylinder. These fins were 0.1 times the diameter of the cylinder in height. In turbulent flows, for reduced velocity (for definition see Table II) ranging between 4 and 6, they observed increased vibrations, as high as five times that observed with a smooth cylinder.

In summary, VIV is a fascinating coupling between body motion and the vortex shedding process and many questions still remain unanswered, particularly for applications where the design objective is to enhance VIV. Bernitsas and group at the University of Michigan have successfully augmented cylinder vibrations and have published extensively^7,9,11,17 on this topic. Their experimental work includes extensive testing of cylinders with distributed sandpaper strips (Passive Turbulence Control/PTC).^{4,5,7,11,17–20} This approach proved to be effective in augmenting fluid induced motions as they observed galloping oscillations at high reduced velocities. Different types of galloping responses, hard galloping (requiring a threshold amplitude) and soft galloping (not requiring a threshold amplitude), were observed in their experiments depending upon the position of the rough strip.¹⁷ Majority of the experimental work carried out by the University of Michigan group spans the Transition in shear layer (TrSL3) Reynolds number regime (2–4 × 10⁴< Re < 1–2 × 10⁵) with galloping oscillation observed completely in TrSL3. The response of a circular cylinder, free to vibrate transverse to the flow, depends on the Reynolds number and the surface roughness parameter. However, the flow physics of VIV augmentation has not been explored in the TrSL2 Reynolds number regime (1–2 × 10³< Re < 2–4 × 10⁴) and is the focus of the current experimental work. We report our findings of systemically testing the effectiveness of a wider range of surface protrusion type modifications to a circular cylinder that was aimed at augmenting fluid induced motions in the TrSL2 Reynolds number regime. The rest of the paper is organized as follows. In Sec. II, we discuss the physics of flow around fixed and vibrating cylinders with emphasis on both low amplitude oscillations (VIV) and large amplitude oscillations (galloping). In Sec. III, we discuss the experimental apparatus and the diagnostics that was used for the current studies; followed by Sec. IV where we discuss our findings and explore the power generation potential of such a device based on variations in position of the protrusion, the degree of roughness of the protrusion and the effect of spring stiffness. The various non-dimensional parameters used are listed in Table II; where m is the total moving mass of the system including the effective mass of the spring (kg), $ρ$ is the density of water (kg/m³), D is the diameter of the cylinder (m), L is the length of the cylinder (m), c is the damping constant (N-s/m)of the system, k is the stiffness of the spring used (N/m), m_a is the added mass of the system given by $ma=Camd$⁠; $md=πρD2L/4$ is the mass of the displaced fluid; C_a = 1.0 is a constant for a circular cylinder, U is the free stream velocity (m/s), f is the frequency of vibration (Hz), $fn$ is the natural frequency in water (Hz), $fvo$ is the vortex shedding frequency (Hz), and $ν$ is the kinematic viscosity of water (m²/s). Table III compares the various parameters employed in the current work and the work by the Bernitsas group at University of Michigan.

A much studied feature of flow around bluff bodies in general, and around a circular cylinder in particular is that the transition from laminar to turbulent flow occurs over a series of distinct transition states that takes place over an enormous range of Reynolds numbers (Re).^21–23 The state of the flow disturbed by a cylinder may be fully laminar (L), in either of a series of three transitions (TrW, TrSL, and TrBL), or fully turbulent (T). The list of the flows states and adopted notations is given in Table IV. Typical transition states of flow around the circular cylinder are sketched in Figures 3(a)–3(d). The first transition state occurs in the wake (Fig. 3(a)) where laminar vortices become turbulent due to three-dimensional distortions further downstream of the flow. The turbulence spreads upstream with an increase of Re; however, the free-shear layers surrounding the near wake always remain laminar. The second transition occurs in the free-shear layers (Fig. 3(b)). The transition region gradually moves upstream towards the separation point as Re is increased. The third transition occurs around separation, seen in Fig. 3(c), and produces the largest effect on the drag force as a result of a complicated interaction between the separation and transition before the boundary layer becomes fully turbulent. The final and fourth transition takes place in the boundary layer away from separation (Fig. 3(d)). The transition region moves upstream with an increase in Re and it ultimately leads to a merger of the transition and stagnation points.²² Beyond the fourth transition state, all disturbed regions of the flow are fully turbulent.

Experimental observations over the last century have revealed an enormous variety of regular and irregular flow regimes around circular cylinders. Zdravkovich²³ provides an extensive review of these flow patterns. The flow regimes and its subdivisions are expected to be confined in the fixed range of Reynolds numbers (see Table IV). The variation of flow pattern in these regimes causes a continuous or discontinuous change of fluctuating and time-averaged (mean) forces exerted on the cylinder. There are a total of six forces (or force coefficients) that act on the bluff body: the time averaged and fluctuating drag and lift forces; denoted as $CD,C′D,CL,C′L$⁠. The drag force is further sub-divided into a skin friction drag (⁠$CDf$⁠) caused due to viscous forces along the surface and a pressure drag (⁠$CDp$⁠), resulting due to the asymmetric pressure distribution on the upstream and downstream sides of the cylinder. The variation of all the six force coefficients acting on a nominally two dimensional cylinder over a wide range of Re is presented in Figure 3(e) for a disturbance-free flow (vertical bars are used to represent the scatter in experimental data). A closer look at Fig. 3(e) presents some important flow phenomenon. The viscous friction coefficient (C_Df) is substantial in the laminar state (L). It keeps decreasing with an increase in Re and becomes negligible starting from TrSL3. The pressure-drag (C_Dp) shows a dependence on the different flow regimes. The least values of C_Dp was observed at the end of L2 (steady and closed near wake), at the transition from TrSL1 to TrSL2 (formation of long eddies) and in TrBL2 (formation of separation bubbles on both side). An increase in fluctuating lift coefficient (⁠$C′L$⁠) is seen throughout TrSL2 and the high values are maintained throughout TrSL3. The single bubble regime, TrBL1, is marked by a discontinuous fall in C_D and the appearance of mean C_L. A discontinuous fall in both C_D and C_L is observed at the start of the two bubble regime, TrBL2. $C′L$ is always observed to be greater than $C′D$⁠.

Vortex shedding is a measure of the nature of flow dynamics associated with a vibrating entity and has been widely studied for an oscillating, elastically mounted cylinder.²⁴ To illustrate the physics of VIV, we consider the experimental configuration that has been used in our laboratory (see Figure 4). The configuration is similar to the classical setup used by Khalak and Williamson²⁵ in their VIV experiments. A rigid circular cylinder, free to oscillate in a direction transverse to the free stream is immersed in a flow. The characteristic parameters of the setup are the oscillating mass of the system and the mass of the fluid displaced by the immersed segment of the cylinder. As the free stream velocity increases, cylinder vibrations develop and reach amplitudes of one diameter over a limited range of velocities associated with a significant change in vibration frequency. Typically, the response profile of a cylinder undergoing VIV is categorized into three regimes, referred to as branches. The initial branch is the regime where the vibrations begin to develop. The upper branch is the region where vibrations with amplitudes of the order of a diameter are observed. In this branch, the vibration frequency tends to synchronize itself to the natural frequency of vibration in water, a phenomena commonly referred to as the lock-in. In the third regime, lower branch, the frequency deviates from the natural frequency and lower vibration amplitudes are observed. At the end of the lower branch, vibrations die down and the region of negligible vibrations that follows is called the desynchronizing regime. Wake patterns induced by the vibration of the body can be classified into three primary modes, namely, the 2S, 2P, and P + S modes, where S stands for a single vortex and P for a pair of vortices. In the 2S mode (2 single vortices), there are two counter rotating vortices shed for one oscillation of the vibrating body (like the classic Karman vortex street). In the 2P mode (2 pair vortices), there are two pairs of counter rotating vortices shed per cycle of the vibrating body. Forced vibration can also lead to other vortex modes including the P + S mode (1 pair + 1 single vortex), which is not able to excite a body into free vibration. Other modes such as C(P + S), C(2S) 2P + 2S, and C (coalescence) also exist²⁶ (see Figure 5, where A is the amplitude of vibration, D is the diameter of the cylinder used, and λ is wavelength of the trajectory along which the body travels relative to the fluid).

Galloping is a flow induced motion which arises due to the asymmetry of the bluff body immersed in the flow and is characterized by high amplitude oscillations.²⁷ Unlike VIV, where the vibration amplitude is self-limited, the vibration amplitude in galloping is self-excited and may keep increasing until system failure occurs unless the vibration is externally restrained. However, galloping does share certain similarities with VIV as it occurs as transverse vibrations in a flow normal to body span with nonlinear oscillations due to the interaction of the wake with the section after body. A slight change in cross section of the bluff body can lead to significant variations in C_D and C_L.²⁸ For example, a deviation from the circular cross-section of transmission lines due to ice formation^29,30 or marine cables due to marine organisms³¹ could cause instabilities and cause galloping. Since asymmetry is essential in inciting galloping, it is never observed in case of a smooth circular cylinder.³² 

Experiments reported in this work were performed using a Water Tunnel facility (Model No. 1520 Rolling Hills Research, CA) with a free surface test section that was 0.381 m wide, 0.508 m deep, and 1.524 m long. The maximum attainable flow velocity in the test section was 0.91 m/s, but all experiments were performed within flow velocities ranging from 0.0569 m/s to 0.5842 m/s. The experimental setup was built using an aluminum frame, a Plexiglas traversing plate, air bearings, mounting blocks, steel rods, an indicator pin, custom made extension springs, and the test cylinder. Figure 4 shows the schematic of the setup. The test cylinder was made of PVC pipe with acrylic end caps and had a length of 0.508 m and a diameter of 0.04826 m; thus the blockage ratio (diameter of the cylinder/width of the channel) was 12.66%. Strips with prescribed roughness (sandpaper and smooth strips) were used as surface protrusion mechanisms for altering the flow around the cylinder. Three sand-paper strips of varying grit size (International Organization for Standardization/Federation of European Produces of Abrasives (ISO/FEPA) designations) were used; the details are provided in Table V. The average grain size refers to the average size of the abrading material, the base size refers to the thickness of the strip on which the abrading material is embedded and the total size is the sum of the base size and the average grain size. Air bearings (Newway, Inc.) were used to allow transverse oscillations at minimal frictional damping and an operating pressure of 80 psi was maintained for all experiments. Stainless steel rods having a diameter tolerance of −0.0005 in. to −0.001 in. were used to mate with the bearing and allow transverse motion. Four pairs of extension springs (W.B. Jones Springs, Inc.), with stiffness of 84.06 N/m, 45.4 N/m, 6.3 N/m, and 1.015 N/m, made of music wire were used for the experiments. The two ends of the spring were hooked to a 6.35 mm (¼ in.) screw-in hook. One hook was bolted to the traversing plate while the other was bolted to a 90 deg bent perforated plate with evenly spaced holes allowing adjustable spring extension. The different configurations experimented are listed in Table VI. All experiments were conducted in the TrSL2 regime, with the Reynolds number varying between 3 × 10² and 3 × 10³. A 1.524 mm (0.06 in.) diameter tracker pin was mounted on the traversing plate; hence, it had a displacement profile identical to that of the cylinder. A high speed camera, Photron FASTCAMX-1024 PCI was tuned to capture the oscillation of the tracker pin. The camera allows capture up to 2.5 × 10⁶ frames/s; however, for the current study, a rate of 60 frames/s was found adequate. The images obtained were intensity images, hence all pixel values ranged between 0 and 1, with black having a value of 0 and white having a value of 1. A MATLAB script, using the image processing toolbox, was used to read and process all the images recorded by the high speed camera. A clear white background was placed behind the indicator pin to provide sufficient contrast between the dark pixels of the indicator pin and the white pixels of the background. This proved highly effective in accurately finding the position of the cylinder for every image read using a sorting algorithm. The script reads all the images taken during an experiment and plots the displacement vs. time graph of the cylinder. The amplitude values plotted are the mean of the top 10% of the amplitudes obtained during an experiment. To obtain the frequency of oscillation, an FFT (Fast Fourier Transform) is performed on the displacement-time values.

Damping tests were conducted to obtain the damping ratio (ζ) of the system. These tests were performed in air and the corresponding damping values, with the different springs used are presented in Table VII. In order to perform the test, the cylinder was given an initial displacement (externally) from the mean position. Owing to damping, the vibrations thus incurred would eventually die down and the cylinder would come back to the mean position. A MATLAB script was used to read and process the captured images. For the springs used in the experiments, the restoring force was assumed to vary linearly with the displacement (effects of nonlinearity have not been considered in the current work). The following expressions were used to calculate the damping ratio (ζ) and the damping constant (c) of the system as:

where δ is the logarithmic decrement and x is the peak displacement from the mean (m). Five damping tests were conducted independently and the values of c and ζ were averaged over the five runs to obtain the final values.

Experiments were performed by placing the different strips of varying roughness (listed in Table V) at different angular positions with respect to the frontal stagnation point. In addition, the effect of spring stiffness on the response of the resultant fluid induced motion was also evaluated using springs of different stiffness. A list of the experiments reported in the current work is listed in Table VI. We discuss our results and the associated flow physics in the following sub-sections to decouple the effect of different parameters on the vibration signature.

In order to study the effect of the position of the strip on cylinder oscillations, experiments were performed by placing P60 sand paper strips at different positions. The angle of separation for the flow around a circular cylinder has been widely studied. It is known to be oscillatory in nature and varies between 75° and 91.5° (measured with respect to the frontal stagnation point) for the flow Reynolds numbers (Re) ranging between 10⁴ and 10⁷.^33–38 For the current experiments, three different angular positions were selected to place the roughness strip at: (1) before the region of natural separation (at 60°); (2) within the region of natural separation (at 80°); and (3) beyond the region of natural separation (at 100°). Figures 6(a) and 6(b) plot the amplitude and frequency response profiles observed while testing different positions for the strip. High amplitude oscillations were obtained for the configuration with strips placed at 60°. The oscillatory nature of flow separation, in case of a smooth cylinder, would imply that location of the point of flow separation would vary along the length of cylinder, leading to a disordered separation line. However, in case of the cylinder with strips at 60°, the front edge of the strips artificially trip separation in a controlled manner, as they are ahead of the region within which the natural angle of separation oscillates. This leads to the straightening of the separation line, increasing the span wise correlation and lift forces, consequently amplifying the vibrations.^11,18 With strips at 60°, galloping oscillations with amplitudes, A* > 1.5, were observed at higher free stream velocities and this region was named as the galloping branch. In the current work, such oscillations were observed beyond a reduced velocity (U*) of 30. In case of the configuration with strips at 80°, a small drop in the maximum vibration amplitude, compared to the smooth cylinder, was observed. However a significant expansion in the range of synchronization was conspicuous. The configuration with strips at 100° proved to be more effective in suppressing the vibrations. The vibration amplitudes and the synchronization range observed for this configuration was smaller compared to the smooth cylinder. A variation was observed in the frequency response of the cylinder depending on the position of the strip (see Figure 6(b)). The smooth cylinder configuration with strips at 80° and 100° showed increasing values of frequency with respect to the reduced velocity. Highest oscillation frequency was observed for the configuration with the strips at 80°, followed by 100° and then the smooth cylinder. The frequency response of the configuration with the strips at 60°, showed a tendency to initially increase and then curve back downwards, to values close to the natural frequency of vibration in water.

Unlike the phenomena of VIV, incitation of galloping oscillations demands an asymmetric bluff body. The asymmetry developed in case of a circular cylinder with strips attached to outer surface, in a manner similar to the configurations used in the current experiments, can be explained as follows. Figures 7(a)–7(c) represent the cylinder at three different instants of time: (a) during upward motion, (b) at the mean position at start, and (c) during downward motion. At position (b), the cross-section of the cylinder is symmetric with respect to the local direction of flow. During upward motion of the cylinder, due to its upward velocity, the local relative velocity of flow will be as shown in position (a) leading to an asymmetric cross-section with respect to the local flow direction. The same happens during the downward motion of the cylinder as depicted in (c). Hence, it can be said that the cross-section, which is symmetric with respect to the direction of free stream velocity at every instant, is never symmetric with respect to the local flow velocity except at the start, leading to the possibility of galloping instability.

Chang et al.⁷ also performed experiments in the TrSL3 Reynolds number regimes (3 × 10⁴< Re < 1.2 × 10⁵) to study the effect of different positions of the strip on vibrations of circular cylinders. They investigated positions in which the upstream location of the strips was varied from 20° to 64°. A more exhaustive investigation of the effect of position in the TrSL3 regime was performed by Park et al.¹⁷ They varied the position of the leading edge of the strip from 0° to 180° and mapped the variation in response of the cylinder, depending on strip location for the P180 and P60 strips in the PTC to FIM map (Passive Turbulence Control to Fluid Induced Motion). Comparing our results with the P60 strip in TrSL2 to that reported by Park¹⁷ for the P60 strip in TrSL3, we observe some interesting differences, which we believe is a consequence of the difference in the Reynolds number regime of operation. While using a ∼30° wide P60 strip at 60°, we observe a soft galloping response reaching amplitudes as high as A* ∼ 2. Whereas in TrSL3, Park et al.¹⁷ reported hard galloping oscillations with reduced amplitudes (A* ∼ 0.7) with a 16° wide strip at 60°. They attributed this to be a consequence of the overstepping of a 16° wide strip at 60° into the strong suppression zone which reduces vibrations. The differences in the nature of response observed indicate that the effect of position and coverage depend on the Re regime of operation as well and hence may show a different trend in the TrSL2 regime.

Experiments were performed using three different sandpaper strips of ISO/FEPA grit designations listed in Table V. All the experiments reported in this section were performed by placing the strips at an angle of 60° from the stagnation point, as it was found to be the most effective, among tested locations, for enhancing oscillations. Figure 8(a) plots the amplitude response of different cylinder configurations with rough strips along with the response of a smooth cylinder. An increase in oscillation amplitudes can be observed for all three configurations with the roughness strips. Also, a change in the (amplitude) response pattern is evident. The smooth cylinder used, followed the traditional three branch response for a low mass-damping system. The initial branch was observed within $4.0≤U*≤5.5,$ upper branch within $5.5≤U*≤14.5,$ and the lower branch within $14.5≤U*≤31$⁠. When compared to other smooth cylinder vibration experiments,^17,39 the response pattern observed is similar, but there are slight variations in the response amplitudes and synchronization range caused due to differences in the mass ratio and mass damping parameter of the experimental setups. In case of the configurations with the rough strips, the extent of the initial branch was not noticeably altered; however, a significantly extended upper branch, within the range $6≤U*≤30$⁠, was observed. Additionally, the galloping oscillations induced led to the galloping branch for U* > 30, where oscillations with amplitudes as high as A* = 2 were achieved. The oscillation amplitudes of the smooth cylinder and the cylinder with roughness strips were comparable in range of reduced velocities $5≤U*≤12$⁠. For a value of U* > 12, the smooth cylinder enters the lower branch with reduced amplitudes of vibration, whereas the configuration with the rough strips continues to vibrate in the upper branch. While VIV undergoes desynchronization beyond U* ≥ 30 for the smooth cylinder, galloping oscillations initiate for the cylinder with strips. The reduced velocity range over which the galloping oscillations are observed is different from that observed by Park et al.¹⁷ and Chang et al.⁷ This is due to the low natural frequency of the system employed in our experimental work, which leads to amplified values of reduced velocity, especially towards the higher free stream velocities (please see Tables III and VII).

The enhanced oscillations observed while employing the rough strips is clearly an outcome of a significant modification incurred by the flow field. The sandpaper strips, attached at specific locations on the outer surface of the cylinder, serve as a surface protrusion based mechanism of altering the flow field. The thickness of the strip primarily affects the flow separation, whereas the embedded abrasive particles affect the shear layers emanating from the surface by adding vorticity to it. Differences were observed between the responses of the cylinder depending on the roughness of the strip attached to it. Eddies shed from the grits are expected to be same order as the grits⁴⁰ and leads to higher amplitudes obtained using the P36 strip (largest grit size) for the range $14≤U*≤26$⁠. The maximum amplitude obtained in galloping did not show any dependence on the roughness. Figure 8(b) plots the frequency response profile of the configuration with the rough strips. The vibration frequencies of all configurations with rough strips showed the tendency to increase initially and then curve downwards to the naturally frequency of vibration in water.

Experiments were performed by replacing roughness strips with smooth strips/zero roughness strips to investigate the changes in response. By doing so, the ability to affect the vorticity of the shear layer was minimized; but the capability to affect separation was retained. The response profiles are plotted in Figures 9(a) and 9(b). By comparison, it can be seen that a cylinder with the smooth strips does have a lower branch of response following the upper branch, unlike the case of a cylinder with rough strips. The amplitudes of vibration for the cylinder with the smooth strips are consistently low compared to the case of the cylinder with the rough strips until the value of U* ∼ 30. The increased amplitudes for the latter configuration in this region could probably be a consequence of the eddies formed behind the roughness elements which add energy to the boundary layer and the vortices shed, which in turn amplify the fluctuating lift force and amplitudes of vibration.⁴⁰ In the galloping branch (U* > 30), higher values of A/D were obtained for the configuration with the smooth strips as compared to the rough strips. Figure 9(b) compares the frequency response profile of the configurations with rough strips and the smooth strips. The tendency to oscillate at the fundamental frequency is a typical signature of galloping instability. This tendency was observed with all configurations employing rough strips; however, the configuration employing the smooth strips exhibited an atypical behavior by galloping at higher frequencies, hinting towards being a better mechanism of augmenting oscillations and energy extraction. Figure 10 summarizes the discussed response of the smooth cylinder, cylinder with P60 sandpaper strip and cylinder with zero roughness strip individually, showing the different response branches and the Reynolds number regime of operation. Power spectrum plots representative of the different response branches observed with the different configurations are plotted in Figure 11. For the case of the smooth cylinder, a dominant frequency peak is observed throughout the upper branch (Figure 11(i–a)). As the flow velocity further increases, a decrease in the dominance of the frequency peak can be observed in the lower branch (Figure 11(i–b)) finally resulting in desynchronization at a velocity of 0.57 m/s (Figure 11(i–c)). In the case of the cylinder with the rough strips, a dominant frequency peak was observed within the initial half of the upper branch (Figure 11(ii–a)), but not throughout its wide upper branch. The latter half of the upper branch showed less dominant frequency peaks (Figure 11(ii–b)), however, in the galloping branch, dominance of the peak increased at frequencies close to the fundamental oscillation frequency in water (Figure 11(ii-c)). In case of the configuration with the smooth strips, dominant frequency peak can be observed throughout the upper branch (Figure 11(iii-a)), similar to the case of a smooth cylinder. In the lower branch, the dominance of frequency observed gradually decreased to a pattern that resembled desynchronization (Figure 11(iii-b)). In the galloping branch that followed, dominance in peak began to reappear gradually, but interestingly at the second harmonic of the fundamental frequency of oscillation (Figure 11(iii-c)), unlike the case of the configuration with the roughness strips.

Comparing the results obtained with the different strips in the current work to that presented by Chang et al.,⁷ we see that certain response characteristics are preserved irrespective of the Reynolds number regime, but some are not. All tested grades of sandpaper strips were successful in initiating galloping oscillation at high velocities in addition to vortex induced vibration at the lower velocities. With the onset of galloping, the frequency ratio of cylinders with the rough strip showed the tendency to approach values close to unity and the maximum amplitudes while executing galloping oscillations did not show a significant dependence on the roughness height. However, the partial suppression of vibration is the VIV synchronization regime reported with all roughness strips in TrSL3 was not observed (while using the rough strips) in the current experiments in TrSL2. Additionally, comparing the response of a cylinder with smooth strip to that of a cylinder with rough strip in TrSL2, we observe a response pattern that is different from that reported by Chang et al.⁴¹ in TrSL3. We observe a partial suppression of vibrations in the VIV regime but enhanced galloping oscillations while using a smooth strip when compared to a rough sandpaper strip, which was quite the opposite of the observation made in TrSL3.

For a mass-spring-damper system, similar to the one employed in the current work, the equation of motion can be written as

where $ffluid(t)$ is the fluid dynamic force acting on the cylinder, t is time, $y$ is the displacement of the cylinder at that instant, and C_total is the total damping of the system. The total mechanical power of such a system could be estimated to be

If we assume the cylinder response to be sinusoidal,⁴² the contributions due to the $y¨$ term and $y$ term turn out to be zero. The only term retained would be the $ẏ$ term and Eq. (3) simplifies to

Hence, it becomes explicit that the power harnessed by such a system would be proportional to the square of the instantaneous oscillation velocities of the different configurations for a given total damping. In order to qualitatively assess the comparative power extraction potential of the different tested configurations, we take a closer look at their oscillation velocities. Figure 12 plots the square of the root mean square (rms) oscillation velocities (V_rms²) observed while using the different strips on the cylinder as a function of the Reynolds number for k = 6.3 N/m. Because of the direct dependence of the mechanical power on the instantaneous velocities of oscillations (Eq. (4)), it is reasonable to expect that power harnessed by the different configurations would follow a trend that is similar to the V_rms² of the different configurations. Higher the V_rms², higher would be the expected power output for a given value of total system damping. Until a Re of 10 000, which corresponds to the initial and upper branch of a smooth cylinder, the differences observed in the V_rms² of the different configurations are not significant. In the range 10 000 < Re < 20 000, the smooth cylinder vibrations begin to die down as they enter the lower branch and the desynchronization regime, but the configurations with the strips transition from VIV to galloping oscillations as described earlier. Within this range, the V_rms² observed with the rough strips were higher compared to the case with the smooth strips almost until the end and had a pattern very similar to the vibration amplitudes within the corresponding reduced velocity range. Beyond Re ∼ 20 000, the configurations with the strips start undergoing galloping oscillations and a linear increase in V_rms² is observed with Re. Within the galloping branch, the V_rms² observed while using the smooth strips are dramatically higher when compared to the V_rms² observed with the rough strips. This is due to the previously discussed difference in vibration frequency of the cylinder with rough strips and the cylinder with smooth strips. The configuration with the smooth strips reaches higher amplitudes at a higher frequency within the galloping branch. With the rough strips, the V_rms² showed a dependence on the grit size. For a given Re, max V_rms² was observed with P36, followed by P60 and P320.

Based on the measured V_rms² values in our experiment, we can conclude that in Re regimes corresponding to VIV lock-in (synchronization), the power extraction capabilities of the different configurations are not significantly different. However, in regimes beyond the VIV upper branch, the smooth cylinder loses its power extraction capability as its vibrations die down, whereas the configurations with strips present an even higher power extraction capability as they undergo galloping oscillations with higher V_rms² values than observed in the VIV upper branch. Comparing the configurations with strips, the dramatically high V_rms² values observed with the smooth strips, is a clear indication of their enhanced capability in energy extraction while executing galloping oscillations.

Experiments were repeated using springs with different values of stiffness ranging from 1.015 N/m to 84.06 N/m. The primary motivation was to identify stiffness-roughness combinations at which the galloping branch would cease to exist. The amplitude and frequency response profiles showing the effect of stiffness for the case of a cylinder with smooth strips are shown in Figures 13(a) and 13(b). Since a variation in stiffness, caused a variation in the range of U*, A* is plotted as a function of the flow Re. The galloping branch was observed with all the four different springs tested for the smooth strips as shown in Figure 13(a). Since galloping is not a self-limited phenomenon, a cap to the vibration amplitudes must be provided by external means; a possibility might be to use springs with larger stiffness. This would mean that, stiffer the spring, lesser the vibration amplitudes within the galloping branch. This pattern can be clearly seen in Figure 13(a). With the least stiff spring tested (k = 1.015 N/m), the galloping oscillations of the cylinder were large enough to collide with the walls of the water tunnel test section. The vibrational frequencies observed were lower with the less stiff springs. However, as seen from Figure 13(b), the proportion of these frequencies to corresponding natural frequency of the system kept on increasing resulting in high frequency ratios. With the stiffer springs (k = 45.4 N/m and 84.06 N/m), the oscillation frequency during galloping seemed to lock in to the fundamental frequency, whereas for the less stiff springs (k = 1.015 N/m and 6.3 N/m), the oscillating frequency seemed to reach higher harmonics of the fundamental frequency.

Experiments were then repeated with cylinders with distributed roughness. Galloping responses were observed while using the less stiff springs (6.3 N/m and 1.015 N/m). With increased stiffness to 45.4 N/m and 84.06 N/m, it appeared that the rough strips, unlike the smooth strips, could not completely transition the response from VIV to the high amplitude galloping oscillations. Figure 14 shows the amplitude response profiles of the different configurations while using a spring of stiffness 45.4 N/m. Clearly, the high amplitude galloping oscillations observed while using the smooth strips are not observed while using the rough strips. The difference in the response is a consequence of the cross sectional profile of the configurations (all the other governing parameters are held constant), and could imply that higher flow velocities which in turn implies a higher flow Re have to be reached before the configurations with the rough strips start executing the high amplitude galloping oscillations. This hints towards the higher susceptibility of the configurations with smooth strips to galloping oscillations and their potential as a better energy extraction mechanism.

The experimental results presented, clearly display the effectiveness of surface protrusion based features like strips in enhancing cylinder vibrations and augmenting energy extraction in the TrSL2 Reynolds number regime. The main findings are listed below.

• Position of the strip with respect to the stagnation point is of pivotal importance for enhancement. From the tested positions, 60° from the frontal stagnation point proved to be most beneficial. They cause the cylinder to transition naturally from VIV to galloping (soft galloping), leading to even higher amplitude oscillation at flow velocities where VIV dies down. When compared to the results presented in the PTC-to FIM map¹⁷ (developed in the TrSL3 Reynolds number regime), we observe a difference in response, hinting towards the dependence on the Reynolds number regime of operation.

• From the tests with the different grades of rough strips, a slight dependence on the grit size was observed, especially towards the latter half of the upper branch. But the maximum amplitudes obtained while executing galloping oscillations did not show a dependence on the grit size. A similar observation was made in the enhancement experiments in TrSL3.⁷ However, the partial suppression reported in the VIV regime while using the rough strips (in TrSL3) was not observed in our experiments in TrSL2.

• Unlike the observations made in TrSL3,⁴¹ where the rough strips are more effective than the smooth strips, our experiments within TrSL2 indicate the smooth strips to be a more promising mechanism for enhancing vibrations. They led to galloping oscillations with greater amplitude and greater frequency, indicating a better energy extraction potential. The smooth strips incited galloping oscillations (with slightly reduced amplitudes) even while employing stiffer springs. This tendency was not seen in the case of rough strips.

• Based on the V_rms² values plotted for the different cylinder/roughness configurations, it appears that the configurations with the strips are capable of extracting mechanical power, comparable to that of the smooth cylinder within VIV lock in and even higher values of power in regimes where the smooth cylinder undergoes desynchronization. The smooth strips were able to reach dramatically higher values of V_rms² while executing galloping oscillations, hinting towards a higher energy extraction capability.

• From our experiments with springs of different stiffness, it was observed that the cylinders with the smooth strips were able to incite galloping responses for all values of stiffness tested, whereas the cylinders with the rough strips were able to do so only with the less stiff springs (6.3 N/m and 1.015 N/m). With increased stiffness (45.4 N/m and 84.06 N/m), it appeared that the rough strips, unlike the smooth strips, could not completely transition the response from VIV to the high amplitude galloping oscillations.

• With the stiffer springs (k = 45.4 N/m and 84.06 N/m) the oscillation frequency of the cylinder with smooth strips, during galloping seemed to lock in to the fundamental frequency, whereas for the less stiff springs (k = 1.015 N/m and 6.3 N/m), the oscillating frequency seemed to reach higher harmonics of the fundamental frequency.

• The ability of the smooth strips to extract higher energy while executing galloping oscillations, and the capability of inciting galloping even at higher stiffness's make them a more effective means of augmenting energy extraction while operating within the TrSL2 Reynolds number regime.

The authors acknowledge support of the Office of Naval Research and Dr. Ronald Joslin for (Grant No. ONR-000141210495). Both authors would also like to thank Varun Lobo for his help with setting up the experimental facility for this work.