2.1 The set-up

An equilateral triangular prism with sides of $D=16$  mm and a length of $L=292$  mm ( $L/D=18.25$ ) was tested for angles of attack of $\unicode[STIX]{x1D6FC}=0^{\circ }$ – $60^{\circ }$ with increments of $5^{\circ }$ (figure 1 a). The prism was made of aluminium and each side was ground, yielding a smooth surface. Attached to the bottom of the prism was a $115$  mm diameter circular endplate with a thickness of approximately $2$  mm and tapered edges to reduce end effects. The prism was attached to an air bearing set-up, allowed to oscillate in the transverse (cross-flow) direction. The air bearing set-up had two air bearings to reduce the damping and to constrain the oscillations of the prism to one degree of freedom in the cross-flow (CF) direction (figure 1 b). Springs were attached from the air bearing set-up to a fixed housing. This set-up was used previously by Seyed-Aghazadeh, Carlson & Modarres-Sadeghi (Reference Seyed-Aghazadeh, Carlson and Modarres-Sadeghi2015b ) to study VIV of tapered cylinders where the response of a uniform circular cylinder was shown to be in agreement with the results of Khalak & Williamson (Reference Khalak and Williamson1999).

For comparison, a circular cylinder made of aluminium with diameter $D=16$  mm, and the same length, $L=292$  mm, was considered as well. A 200 g dummy mass was used to adjust the mass ratio of the circular cylinder to that of the triangular prism, $m^{\ast }=9.24$ ( $m^{\ast }=m/\unicode[STIX]{x1D70C}AL$ , where $m$ is the total moving mass, $\unicode[STIX]{x1D70C}$ the fluid density, $A$ the cross-sectional area and $L$ the length). All moving parts were accounted for in this calculation, including the dummy mass, air bearings, mounting bracket, force sensor and various cabling.

2.2 Flow quality

The recirculating water tunnel used featured a test section with the dimensions $1.27~\text{m}\times 0.5~\text{m}\times 0.38~\text{m}$ . The flow profiles were measured using bubble image velocimetry (BIV). The cathode to generate hydrogen bubbles was a platinum wire $50~\unicode[STIX]{x03BC}\text{m}$ in diameter, held taut in pure cross-flow via streamlined mounting plates affixed to the test section walls. At 35 mm downstream of the cathode, a carbon plate was submerged at the midpoint of the test section, oriented parallel to flow, and served as the anode. By completing the anode–cathode circuit with a 50 V, 0.25 A DC power source, the voltage passing from the cathode separated hydrogen from the water. Hydrogen bubbles accumulated on the trailing face of the wire until they surpass the wire diameter, at which point they were shed downstream. Thus, a uniform bubble sheet was generated across the width of the test section with a bubble size of approximately $50~\unicode[STIX]{x03BC}\text{m}$ . For viewing purposes, light-emitting diode (LED) lighting banks were mounted at a $45^{\circ }$ angle to the bubble plane to ensure uniform lighting conditions. A high-speed camera facing perpendicular to the cross-flow–streamwise plane captured frames at 120 frames per second. The entire platinum array was traversed vertically to several different horizontal planes to capture the vertical nature of the flow profile.

Velocimetry analysis was done via open-source PIV software, PIVlab (Thielicke & Stamhuis Reference Thielicke and Stamhuis2014). The field was interrogated with two passes: $64\times 64$ pixel down to $32\times 32$ pixel areas with 50 % overlap. The resultant vector field was restricted to values less than three standard deviations above the mean to mitigate numerical error. The flow velocity was measured using the time-averaged values for the vector space directly near the platinum wire. For calculating flow uniformity of the cross-section, 30 mm was cropped from either end where the wire was connected to the wall to exclude the boundary layer effects from this calculation. Measurements were conducted at $30$ , $76$ , $120$ and $170~\text{mm}~\text{s}^{-1}$ , and at each flow velocity at four different heights. Figure 2 shows a sample case of the measurements at a flow velocity of $76~\text{mm}~\text{s}^{-1}$ , based on these measurements, the flow uniformity across the test section was calculated and it was found to be within 2–8 % for the range of flow velocities used in the current experiments.

Figure 2.

The flow profile measured along the width of the test section (the $w$ -direction) at different heights (the $z$ -direction) for a sample flow velocity of $U_{0}=76~\text{mm}~\text{s}^{-1}$ .

2.3 Data collection method

Data were recorded for this experiment via two different sensors. A Micro-Epsilon ILD 1402-600 non-contacting displacement sensor recorded the transverse displacements. This sensor ran in tandem with an ATI-Nano17/IP68 six-axis force sensor that was attached to the upper end of the oscillating prism and measured the total flow forces acting on the prism in three perpendicular directions. This set-up enabled a complete measurement of the lift and drag forces acting on the prism. Decay tests were conducted in both air and water, by giving the prism an initial displacement and then recording the amplitude of oscillations for over 15 cycles of oscillations. Decay tests in water yielded a natural frequency of $f_{nw,T}=0.466\pm 0.007$  Hz and $f_{nw,C}=0.383$  Hz for the triangular prism and circular cylinder, respectively. Decay tests in air resulted in a natural frequency of $f_{na,T}=0.503$  Hz and $f_{na,C}=0.397$  Hz, and a structural damping ratio of $\unicode[STIX]{x1D701}=0.0057$ . The structural damping ratio was measured for different initial displacements of the prism, and resulted in the same value with a standard deviation of order $5\times 10^{-4}$ .

During these experiments, the water level was held constant and the flow velocity was increased from zero in small steps up to a Reynolds number of $Re=2700$ . At each step, the amplitude and frequency of oscillations as well as the flow forces acting on the prism were measured. Experimental runs were recorded over a 60 s duration at each flow velocity and angle of attack. This duration covered over 100 periods of oscillation of the prism.

2.4 Flow force measurements

The force sensor attached to the upper end of the prism measured the total flow forces acting on the prism in both the cross-flow and inline directions. The measured force in the cross-flow direction comprised both the hydrodynamic flow forces and the inertia force due to the motion of the force sensor. Therefore, the inertia force was removed from the total measured transverse force by post-processing the experimental data.

Figure 3.

Flow forces and their corresponding frequency contents measured directly using the force sensors (solid line) and reconstructed using the method discussed in the text (dotted line) for (a,b)  $\unicode[STIX]{x1D6FC}=55^{\circ }$ and $U^{\ast }=10.7$ (moderate-amplitude response), (c,d)  $\unicode[STIX]{x1D6FC}=35^{\circ }$ and $U^{\ast }=16.7$ (low-amplitude response), and (e,f)  $\unicode[STIX]{x1D6FC}=55^{\circ }$ and $U^{\ast }=20.7$ (large-amplitude response). In the plots, $C_{y}=2F/\unicode[STIX]{x1D70C}DU^{2}L$ and $t^{\ast }=tf_{nw}$ .

An alternative method was also used here to obtain the flow forces. Given the time series of the prism’s cross-flow displacement, the cross-flow force can be calculated as

(2.1) $$\begin{eqnarray}F/m=\ddot{y} +2\unicode[STIX]{x1D701}\unicode[STIX]{x1D714}_{n}{\dot{y}}+\unicode[STIX]{x1D714}_{n}^{2}y,\end{eqnarray}$$

where $F$ is the cross-flow force, $m$ the system’s mass (only the structure, with no added mass contribution), $\unicode[STIX]{x1D714}_{n}$ the natural frequency in air, $\unicode[STIX]{x1D701}$ the damping coefficient measured in air (0.0057 for the present experiments) and $y$ the displacement data recorded using the non-contacting laser sensor. This force reconstruction method was previously used by Seyed-Aghazadeh et al. (Reference Seyed-Aghazadeh, Carlson and Modarres-Sadeghi2015b ) to measure the flow forces acting on a tapered cylinder. In that work, the results were shown to match very well the results of the direct measurements using a force sensor and also those of Khalak & Williamson (Reference Khalak and Williamson1999) for a uniform cylinder with a similar mass-damping coefficient.

Here sample cases are given to show how this method works for measurements from a triangular prism. Figure 3 shows a comparison between the flow forces measured directly by the force sensor and those reconstructed using (2.1), for three sample cases with low, moderate and large amplitudes of oscillations. It is observed that the force reconstruction method is capable of reproducing very fine details of the flow forces, as shown both in the time history and in fast Fourier transform (FFT) plots of all three cases. The flow forces reported in this work are the reconstructed values.

5.1 Angle of attack $\unicode[STIX]{x1D6FC}=35^{\circ }$

Figure 10 shows the dimensionless displacement time histories ( $y/D$ , where $D$ is the side of the triangular cross-section), the cross-flow force coefficients, $C_{y}$ , and their corresponding frequency contents (FFT plots) for these sample points. The FFT plots exhibit distinct frequency contents: while the main peak in all of these FFT plots corresponds to a frequency close to the natural frequency of the system in water, small contributions of the second and third harmonics, i.e. with twice and three times the frequency of oscillations, are observed in some cases.

Figure 10.

Sample time histories and FFT plots of the prism’s dimensionless displacement in the cross-flow direction as well as the cross-flow force coefficient together with their corresponding frequency contents for $\unicode[STIX]{x1D6FC}=35^{\circ }$ at three reduced velocities: (a–d)  $U^{\ast }=10.7$ , (e–h)  $U^{\ast }=16.7$ , and (i–l)  $U^{\ast }=20.7$ .

At a reduced velocity of $U^{\ast }=10.7$ (figure 10 a–d), the cross-flow force consists of the main harmonic (at a frequency close to the natural frequency of the system) as well as the second harmonic (twice the natural frequency of the system). This is a sample point in which the amplitude and frequency of oscillations correspond to a VIV response. The first peak in the FFT plot corresponds to the main frequency of oscillations, and the second peak at the second harmonic is due to the asymmetry of the system. Du, Jing & Sun (Reference Du, Jing and Sun2014) showed that, even for a circular cylinder, with a symmetric cross-section, undergoing VIV, two peaks can be observed in the FFT plot of the cross-flow forces, due to the asymmetry in the shedding. For a triangular prism, due to the added geometrical asymmetry with respect to the incoming flow at $\unicode[STIX]{x1D6FC}=35^{\circ }$ , the second peak at a frequency twice the main frequency of oscillations has an even larger contribution, comparable with the first harmonic frequency. This is in fact due to the contribution of the fluctuating drag force at a frequency twice the fundamental frequency.

For the reduced velocity of $U^{\ast }=16.7$ , the cross-flow force has a very small value (figure 10 e–h). The amplitude of oscillations is also very small at this reduced velocity. Two small peaks are observed in the FFT plot of this reduced velocity: the first peak corresponds to the main frequency of oscillations and the second peak corresponds to the Strouhal frequency. The synchronization between the two frequencies is lost in this region, and therefore no large-amplitude oscillations are observed.

For a larger reduced velocity of $U^{\ast }=20.7$ , the cross-flow force consists of the main harmonic as well as contributions from higher harmonics, i.e. the second, third and fourth harmonics (figure 10 i–l). This sample point falls within the reduced velocity range in which the amplitude of oscillations resembles a galloping-type response.

5.2 Angle of attack $\unicode[STIX]{x1D6FC}=55^{\circ }$

For an angle of attack of $\unicode[STIX]{x1D6FC}=55^{\circ }$ , at a reduced velocity of $U^{\ast }=10.7$ , the cross-flow force mainly consists of the first and the second harmonics, and a slight contribution of the third harmonic is also observed (figure 11 a–d). These peaks correspond to the $\text{T}+\text{P}$ vortex shedding pattern observed at this point, as discussed later in § 7.2. At higher reduced velocities of $U^{\ast }=16.7$ (figure 11 e–h) and $U^{\ast }=20.7$ (figure 11 i–l), large higher harmonic contributions (up to the fifth) in the cross-flow force are observed, which correspond to the high-frequency wake modes observed at these reduced velocities: 2T and $5\text{S}+4\text{S}$ , respectively, again as discussed later in § 7.2. This suggests that the shedding of vortices is independent of the oscillation frequency at these reduced velocities. Both of these reduced velocities correspond to the range of large-amplitude galloping-type response of the system.

Figure 11.

Sample time histories and FFT plots of the prism’s dimensionless displacement in the cross-flow direction as well as the cross-flow force coefficient together with their corresponding frequency contents for $\unicode[STIX]{x1D6FC}=55^{\circ }$ at three reduced velocities: (a–d)  $U^{\ast }=10.7$ , (e–h)  $U^{\ast }=16.7$ , and (i–l)  $U^{\ast }=20.7$ .

Figure 12.

(a) The dimensionless amplitude of the cross-flow oscillations, $A^{\ast }$ , and (b) the dimensionless frequency (circles, $f_{1}$ ; squares, $f_{2}$ ; triangles, $f_{3}$ ) of the cross-flow oscillations, $f^{\ast }=f_{osc}/f_{na,T}$ , versus the reduced velocity, $U^{\ast }=U/f_{nw,T}D$ . Contour plots of the dimensionless frequency of (c) the cross-flow oscillations, (d) the total transverse force, and (e) the transverse vortex force. All are measured at an angle of attack of $\unicode[STIX]{x1D6FC}=35^{\circ }$ .

Figure 13.

(a) The dimensionless amplitude of the cross-flow oscillations, $A^{\ast }$ , and (b) the dimensionless frequency (circles, $f_{1}$ ; squares, $f_{2}$ ; triangles, $f_{3}$ ; diamonds, $f_{4}$ ) of the cross-flow oscillations, $f^{\ast }=f_{osc}/f_{na,T}$ versus the reduced velocity, $U^{\ast }=U/f_{nw,T}D$ . Contour plots of the dimensionless frequency of (c) the cross-flow oscillations, (d) the total transverse force, and (e) the transverse vortex force. All are measured at an angle of attack of $\unicode[STIX]{x1D6FC}=55^{\circ }$ .

6.1 Angle of attack $\unicode[STIX]{x1D6FC}=35^{\circ }$

At $\unicode[STIX]{x1D6FC}=35^{\circ }$ (figure 12), representative of a ‘distinct VIV–galloping’ trend, all three types of response, i.e. VIV type, galloping type and no oscillation, have been observed. Besides the dominant frequencies of oscillations, $f_{1}$ , the higher frequencies at twice and three times the dominant frequency, $f_{2}$ and $f_{3}$ , are also observed. For this angle, the frequency of oscillations (figure 12 b) follows the Strouhal line ( $St$ line in figure 12) for the lower reduced velocities ( $U^{\ast }\sim 4.0{-}4.7$ ), and then locks into the natural frequency of the system at $U^{\ast }\sim 4.7$ and stays close to the natural frequency up to $U^{\ast }=7.4$ . In this range of reduced velocities, the response is very similar to a classic VIV response when a $1:1$ ratio between the shedding frequency and the oscillation frequency triggers VIV. At $U^{\ast }=7.4$ , there is a slight decrease in the amplitude of oscillations, which is accompanied by a drop in the frequency of oscillations from $f^{\ast }=0.93$ to $f^{\ast }=0.63$ . Then for the reduced velocity range of $U^{\ast }=7.4{-}14.2$ , the frequency of oscillations initially follows the line corresponding to half the Strouhal frequency ( $St$ /2 line in figure 12) for $U^{\ast }\sim 7.4{-}10.1$ until it reaches a value close to the natural frequency of the system again at $U^{\ast }\sim 10.1$ . In this range, the synchronization occurs between the system’s natural frequency and half the shedding frequency calculated based on $St$ . At this point, the frequency once again locks into the natural frequency of the system. The dimensionless frequency stays around unity up to a reduced velocity of $U^{\ast }\sim 14.2$ , beyond which the prism can no longer stay locked into the natural frequency of the system and the amplitude of oscillations decreases suddenly. Higher harmonics at twice and three times the dominant frequency of oscillations are also observed in this range of reduced velocities ( $U^{\ast }=7.4{-}14.2$ ).

Overall, the response of the system in the frequency domain in the reduced velocity range of $U^{\ast }=4.0{-}14.2$ resembles that observed in the lock-in response of a VIV case, in the sense that non-zero-amplitude responses are observed over a range of reduced velocities, being preceded and followed by zero-amplitude responses. In the case of a triangular prism, however, the frequency of oscillations locks into the natural frequency twice: once approaching from the Strouhal frequencies and once from half the Strouhal frequencies. Another difference compared with the classic VIV response is that the dimensionless frequency never crosses one (figure 8), thus resulting in no jump in the phase difference between the structure’s displacement and flow forces.

In the reduced velocity range of $U^{\ast }=14.7{-}19.4$ , where the dimensionless amplitudes of oscillations stay at small values of less than $A^{\ast }=0.1$ , the frequency content has a contribution from the natural frequency of the system and a contribution from the shedding frequency at a value following the Strouhal frequency of a fixed prism at the same angle of attack ( $St$ line in figure 12). The shedding frequency increases linearly with reduced velocity, as expected, but the contribution from the natural frequency of the system stays constant. A range of broad-band, low-power frequencies, following the Strouhal line, is also observed in the transverse vortex forces of figure 12(e).

At a reduced velocity of $U^{\ast }=19.6$ , where the frequency component from the Strouhal line gets close to three times the natural frequency of the system, the prism starts to oscillate again. It seems that the matching of the shedding frequency and three times the natural frequency acts as a trigger for the system to restart its oscillations at this reduced velocity. The dominant frequency of oscillations is close to the natural frequency of the system. Higher harmonic components at twice and three times the dominant frequency also exist in this range. The second harmonic does exist due to the asymmetry in shedding as discussed before in § 5.1. The third harmonic component is mainly due to the relative angle of the lift and drag force with respect to the direction of the prism’s oscillation. Owing to this relative angle, the flow force in the direction of oscillation comprises components of lift and drag forces, at the first and second harmonics, which give rise to forces at three times the fundamental frequency. Seyed-Aghazadeh, Budz & Modarres-Sadeghi (Reference Seyed-Aghazadeh, Budz and Modarres-Sadeghi2015a ) discuss more details of the third harmonic’s contribution. In the frequency contents of the transverse vortex force of figure 12(e), peaks at the first, second and third harmonic components are observed, but the second harmonic is the dominant frequency.

6.2 Angle of attack $\unicode[STIX]{x1D6FC}=55^{\circ }$

Figure 13 shows the same plots as figure 12, but for an angle of attack $\unicode[STIX]{x1D6FC}=55^{\circ }$ . The oscillations start at a frequency close to the Strouhal frequency at a reduced velocity of $U^{\ast }=5.4$ . The frequency of oscillations follows the Strouhal line (the continuous line in figure 13 b) and, as the shedding frequency gets closer to the natural frequency of the system, it locks into the natural frequency. The frequency of transverse vortex force (figure 13 e) stays close to the frequency of oscillations for a range of reduced velocities, resulting in a $1:1$ synchronization in this range, similar to what was observed for the VIV range of the response for $\unicode[STIX]{x1D6FC}=35^{\circ }$ . The reduced velocity of $U^{\ast }=5.4$ , therefore, can be considered as the onset of the VIV response for the $\unicode[STIX]{x1D6FC}=55^{\circ }$ case, for which the frequency of vortex shedding gets locked into the frequency of oscillations.

The ‘lock-in’ region seems to be extending up to around $U^{\ast }\sim 10$ , where a sudden change in the slope of the amplitude plot is observed. This ‘kink’ corresponds to the reduced velocity at which the dominant frequency of the transverse vortex force shifts from a value equal to that of the oscillation frequency to twice the oscillation frequency. This jump in the dominant frequency implies that the vortex shedding frequency is no longer ‘locked into’ the oscillation frequency of the prism, and therefore the reduced velocity at which this jump occurs (i.e. $U^{\ast }\sim 10$ ) can be considered as the end of the lock-in range and therefore the end of the VIV response. At reduced velocities following this range, the vortex shedding is synchronized to twice the oscillation frequency, resulting in a $1:2$ synchronization.

For $U^{\ast }\sim 10{-}15$ , the dominant dimensionless frequency of oscillations stays around unity, while the transverse vortex force is dominated by a frequency twice the oscillation frequency. The higher harmonic components of the oscillation frequency also exist at twice, three times and four times the dominant frequency. The amplitude of oscillations increases monotonically by increasing the reduced velocity in this range. At $U^{\ast }\sim 15$ , another sudden change in the slope of the amplitude plot is observed (a ‘kink’). This change occurs at a reduced velocity for which the first harmonic of the transverse vortex force becomes large again and its values become comparable to its higher harmonics.

For $U^{\ast }\sim 15{-}20$ , the amplitude of oscillations increases monotonically with increasing reduced velocity and the second harmonic of the transverse vortex force remains the dominant frequency. At $U^{\ast }\sim 20$ , where the third harmonic frequency matches the vortex shedding frequency of a fixed prism, the third ‘kink’ in the amplitude of oscillations is observed. Beyond this reduced velocity, the second and the third harmonic transverse vortex forces become broad-band and the amplitude of oscillations increases with increasing $U^{\ast }$ . The values of the third harmonic vortex lift force become comparable with the dominant second harmonic frequency, resulting in a combined $1:2$ and $1:3$ synchronization.

Zhao et al. (Reference Zhao, Leontini, Lo Jacono and Sheridan2014) observed similar behaviour for a low-mass-ratio square prism at an angle of attack of $\unicode[STIX]{x1D6FC}=0^{\circ }$ , where the flat face of the square was placed perpendicular to the flow. Different synchronization regimes have been reported in their work, where vortex shedding was being locked into multiple integers of the body oscillation frequencies. The observed synchronization at $\unicode[STIX]{x1D6FC}=0^{\circ }$ was only at odd multiple integers of the vortex shedding cycles, referred to as ‘odd’ synchronization. No ‘even’ synchronization was observed for the square prism at $\unicode[STIX]{x1D6FC}=0^{\circ }$ . It was concluded by the authors that for a square prism placed in flow in a symmetric orientation, such as $\unicode[STIX]{x1D6FC}=0^{\circ }$ , even vortex shedding produces a mean lift force of zero, which cannot lead to a significant transverse galloping response. Based on the findings of the present study, however, it seems that we do not necessarily observe an odd synchronization if the body is not placed symmetrically in the flow. The body’s asymmetric orientation can cause asymmetric vortex shedding (in terms of the size and strength of the vortices) from each side of the prism, thus resulting in a non-zero mean lift. Since the number of shed vortices from each side is the same, this results in an ‘even’ synchronization, but because the vortices are not necessarily of the same size and strength, they could produce non-zero mean lift. This matches the behaviour that was observed in the two angles of attack considered here, where for both $\unicode[STIX]{x1D6FC}=35^{\circ }$ and $\unicode[STIX]{x1D6FC}=55^{\circ }$ the prism was oriented asymmetrically with respect to the incoming flow, and therefore both odd and even synchronizations were observed in these two cases.

To investigate if we observe only ‘odd’ synchronization in a symmetrically oriented triangular prism, the angle of attack of $\unicode[STIX]{x1D6FC}=60^{\circ }$ was considered. Figure 14 shows the contour plots of dimensionless frequency of oscillations, total transverse force and vortex transverse force frequencies for the angle of attack of $\unicode[STIX]{x1D6FC}=60^{\circ }$ , where the triangular prism is placed symmetrically with respect to the incoming flow. In this case, the dimensionless frequency of oscillations stays around unity for the entire range of reduced velocities tested. However, in addition to the main harmonic frequency, there is a large contribution from the third harmonic in the total transverse force contour. Considering only the vortex shedding component of the transverse force, the third harmonic is the dominant frequency, and the contribution from the second harmonic is minimal, implying a $1:3$ synchronization. This ‘odd’ synchronization, as a consequence of symmetric vortex shedding in the wake of a triangular prism placed symmetrically with respect to the flow, agrees with the results observed by Zhao et al. (Reference Zhao, Leontini, Lo Jacono and Sheridan2014), where they did not observe any contribution from even harmonics in the flow-induced oscillations of a square prism placed symmetrically in flow.

Figure 14.

Contour plots of the dimensionless frequency of (a) the cross-flow oscillations, (b) the total transverse force, and (c) the transverse vortex force. All are measured at an angle of attack of $\unicode[STIX]{x1D6FC}=60^{\circ }$ .

6.3 Phase differences

Figure 15 shows how the phase difference between the first harmonic displacement and the first harmonic cross-flow force, $\unicode[STIX]{x1D719}$ , changes with the reduced velocity for two angles of attack $\unicode[STIX]{x1D6FC}=35^{\circ }$ and $55^{\circ }$ . The phase difference for both cases stays around $10^{\circ }$ for all reduced velocities, except for a small range of higher reduced velocities for $\unicode[STIX]{x1D6FC}=35^{\circ }$ , where the phase difference increases to values close to $30^{\circ }$ . In all these cases, the phase difference never increases to values larger than $90^{\circ }$ . This behaviour is different from what is observed in the VIV response of a circular cylinder, for which there is a jump from $0^{\circ }$ to $180^{\circ }$ in phasing between flow forces and the cylinder displacement (Khalak & Williamson Reference Khalak and Williamson1999). This phase jump is also observed in the present results for the uniform circular cylinder at a reduced velocity of $U^{\ast }\sim 7$ (circles in figure 15). The maximum value of the phase difference at larger reduced velocities in the current experiment is approximately $130^{\circ }$ , instead of $180^{\circ }$ . This is due to the fact that, with a non-zero structural damping, the phase jump to $180^{\circ }$ occurs only if the frequency ratio ( $f_{osc}/f_{na,C}$ ) gets far from unity. In the current experiments, however, the frequency ratio stayed close to unity, resulting in a phase difference of less than $180^{\circ }$ . At any rate, there is a phase change from small values to values larger than $90^{\circ }$ for the circular cylinder tested here, which means that the flow forces start opposing the displacement at a certain reduced velocity. This phase jump does not occur for the triangular prism. The fact that $f_{osc}/f_{na,T}$ stays at values smaller than one in figure 8 also confirms the monotonic phase change observed here for the triangular prism, whereas the frequency ratio passed unity for the circular cylinder at a reduced velocity of $U^{\ast }\sim 7$ in figure 8.

Figure 15.

The phase difference between the cross-flow amplitude and the cross-flow force.

7.1 The wake at sample points for $\unicode[STIX]{x1D6FC}=35^{\circ }$

Figure 16 shows the vortex shedding patterns in the wake of the prism at the maxima of one oscillation cycle at sample reduced velocities of $U^{\ast }=10.7$ , 16.7 and 20.7. The snapshots on the left are directly from the dye flow visualization images and the schematics on the right specify the observed patterns (also seen inside the boxes drawn around the shed vortices). At the reduced velocity of $U^{\ast }=10.7$ (figure 16 a), two single vortices are shed in each cycle of oscillations (2S pattern). The vortex shedding frequency at this point is equal to the frequency of oscillations ( $f_{vs}\sim f_{os}=0.35$  Hz at $U^{\ast }=10.7$ ), which is a feature of a VIV response.

Figure 16.

Vortex shedding patterns in the wake of the triangular prism at an angle of attack of $\unicode[STIX]{x1D6FC}=35^{\circ }$ at reduced velocities of (a)  $U^{\ast }=10.7$ , (b)  $U^{\ast }=16.7$ and (c–d)  $U^{\ast }=20.7$ .

At $U^{\ast }=16.7$ , the 2S vortex shedding persists (figure 16 b) but the shedding frequency ( $f_{vs}\sim 1.25$  Hz) no longer follows the natural frequency of the system ( $f_{nw,T}=0.466$  Hz), but is equal to the Strouhal frequency for a stationary triangular prism ( $f_{st}=1.28$  Hz) based on the results of figure 5. This was expected, since the amplitude of oscillations in this range is very small (close to zero) and the triangular prism is practically stationary.

At $U^{\ast }=20.7$ , the response is of a galloping type as we discussed in § 4. As shown in figure 16(c,d), at this point five vortices are shed from one side when the prism is moving up and three vortices are shed from the other side of the prism when it is moving down. We call this pattern a $\text{5S+3S}$ pattern, for five single vortices shed in one half-cycle and three single vortices shed in the other half-cycle. Clearly the shedding is asymmetric and its frequency is much higher than the oscillation frequency. The fact that the shedding frequency is much higher than the oscillation frequency suggests that the oscillations are low-frequency and the shedding is a consequence of the presence of the bluff body in the flow and not the forcing that causes the oscillations.

In summary, for this angle of attack, a branch of VIV-type response is followed by a range of almost-zero-amplitude response, and then a branch of galloping-type response at higher reduced velocities. In the VIV-type range of the response, 2S shedding, typical of a VIV response, is observed, while in the galloping-type range of the response, the shedding is asymmetric with a higher frequency compared with the oscillation frequency.

7.2 The wake at sample points for $\unicode[STIX]{x1D6FC}=55^{\circ }$

The second series of wake visualizations discussed here were conducted at the same three reduced velocities of $U^{\ast }=10.7$ , 16.7 and 20.7, but at a larger angle of attack of $\unicode[STIX]{x1D6FC}=55^{\circ }$ (figure 17). At $U^{\ast }=10.7$ , a $\text{T}+\text{P}$ pattern is observed in which a triplet of vortices is shed when the prism moves up and a pair of vortices is shed when the prism moves down (figure 17 a,b). At $U^{\ast }=16.7$ , a 2T pattern (two triplets of vortices shed in each cycle of oscillations) is observed (figure 17 c). The $\text{T}+\text{P}$ and 2T modes have been observed in the VIV response of a circular cylinder as well (Williamson & Govardhan Reference Williamson and Govardhan2004), and, since they have been observed previously, we use symbols T and P for them. With the nomenclature used in this paper, we could call the $\text{T}+\text{P}$ pattern a $3\text{S}+2\text{S}$ pattern, and the 2T pattern a $3\text{S}+3\text{S}$ pattern, as well. At $U^{\ast }=20.7$ , a $5\text{S}+4\text{S}$ pattern is observed (figure 17 d,e) in which five vortices are shed from one side when the prism is moving up and four vortices are shed from the other side of the prism when it is moving down. The $5\text{S}+4\text{S}$ could also be called a 2P, $\text{S}+\text{P}$ , and so on for all the high-frequency multiple shedding patterns. However, to keep the nomenclature simpler, we use $n\text{S}+m\text{S}$ in referring to these shedding patterns, with the understanding that in fact all we do is to count the number of vortices shed from each side of the prism in each cycle of oscillations.

Figure 17.

Vortex shedding patterns in the wake of the triangular prism at an angle of attack of $\unicode[STIX]{x1D6FC}=55^{\circ }$ and at reduced velocities of (a–b)  $U^{\ast }=10.7$ , (c)  $U^{\ast }=16.7$ and (d–e)  $U^{\ast }=20.7$ .

It can be argued that, for this angle of attack, at some reduced velocity, the shedding frequency is so much larger than the oscillation frequency that the response can be considered to be a galloping type as opposed to a VIV type. That would be a galloping-type response as a low-frequency response, where ‘low’ is in comparison with the shedding frequency, which implies that the shedding is the result of the presence of the bluff body in flow and not the cause of its oscillations. The low-frequency oscillations combined with the large-amplitude oscillations (much larger than the typical one-diameter amplitude observed in the VIV response of bluff bodies) indicates a galloping-type response at higher reduced velocities. Overall, for this angle of attack, the initial VIV-type response gives way to a galloping-type response at higher reduced velocities, without any intermediate, almost-zero-amplitude response, which we observed for the angle of attack of $\unicode[STIX]{x1D6FC}=35^{\circ }$ .

7.3 An overall view of the wake

A summary of the shedding patterns observed in the wake of the triangular prism is given in figure 18. The predominant shedding patterns at lower reduced velocities are 2S, $\text{P}+\text{S}$ , 2P, $\text{T}+\text{S}$ , $\text{T}+\text{P}$ and 2T, which have been observed previously for the VIV response of a circular cylinder with various mass-damping parameters. For the cases with a ‘combined VIV–galloping’ trend, at reduced velocities of higher than $U^{\ast }=15$ the shedding frequency becomes much larger than the oscillation frequency, a sign of the onset of galloping. These high-frequency shedding patterns (shown as $n\text{S}+m\text{S}$ in figure 18) are asymmetric ( $n\neq m$ ) for all angles of attack, except for $\unicode[STIX]{x1D6FC}=60^{\circ }$ . This suggests that the asymmetry in the number of shed vortices from each side is only due to the geometric asymmetry of the structure in these angles of attack. In figure 18, the shedding patterns are identified for each angle of attack, and based on the flow visualizations conducted at selected reduced velocities only. Therefore the vertical lines in the plot should not be interpreted as a sudden change of the shedding pattern.

Figure 18.

(a) A map of the vortex patterns in the wake of the triangular prism at varying angles of attack and at different reduced velocities, together with (b) schematics of the shedding patterns.