Effects of Span Length and Additional Structure on Flow-Induced Transverse Vibration Characteristic of a Cantilevered Rectangular Prism

We consider the effects of the aspect ratio L/H (where L is the length of a prism, and H is the height of a prism normal to the flow direction) and the size of additional structures (which are a plate and a fin on the surface of a prism) on a vibration characteristic of a cantilevered rectangular prism. The present research is intended to support the analysis of energy harvesting research on the flow-induced vibration in water flow using a magnetostrictive phenomenon. The prisms are constructed from stainless steel and mounted elastically to a plate spring attached to the ceiling wall of the water tunnel. The prisms with aspect ratios of L/H ≥ 5 have reasonably identical vibration characteristics. However, the difference in the vibration characteristic appears distinctly on a rectangular prism with an aspect ratio of L/H = 2.5. The rectangular prism with an aspect ratio of L/H = 10 and a side ratio of D/H = 0.2 has a stable and large response amplitude and oscillates with a lower velocity. The length of the added plate and the size of the added fin influence the velocity of vibration onset. If the length of the added plate and fin size on the rectangular prism with D/H = 0.2 becomes large, the curve of the response amplitude shifts to that of the rectangular prism with D/H= 0.5. The response amplitude of the rectangular prism with/without plate or fin is found to be related to the second moment of area of the prism.


Introduction
Key features of a two-dimensional prism with a rectangular cross-section in a uniform flow are the generation of alternating vortices behind the prism and the separation bubble on the side surfaces of the prism. Flow characteristics of fluid forces and vortex shedding frequency have been dramatically changed by the side ratio of rectangular prisms [1]. Because of these features, the elastically mounted rectangular prisms experience the flow-induced vibration in crossflow directions by vortex-induced and galloping vibrations [2] [3]. The dynamic response of a prism depends on parameters such as the system damping and after body shape, that is, the structural part of a bluff body downstream from the flow separation points [4].
The flow around a finite-length prism without an end plate becomes a strongly three-dimensional structure [4]- [9]. There are distinct flow pattern changes along the prism height from the tip to the root. The near wake structure is dependent on the aspect ratio, L/H, of the prism, where L is the length of the prism and H is the height of the prism normal to the flow direction. From those investigations, the flow structure behind a finite prism is qualitatively identical to that of prisms above a critical aspect ratio, that is, L/H ≥ 5. The wake structure is characterized by the trailing vortex from the tip of the prism and by the Kármán vortex shedding from the sides of the prism. On the other hand, below the critical aspect ratio, the arch and tip vortices can be observed in the wake flow of a finite prism.
The flow-induced vibration of a cantilever rectangular prism has been investigated by a few researchers [10] [11] [12]. Kiwata et al. [12] investigated the effect of side ratio on the response amplitude characteristic of cantilevered rectangular and D-section prisms with an aspect ratio of 10 and a side ratio of less than D/H = 0.6, where D is the depth of a bluff body in the flow direction. However, it can be inferred that the length of the prism has non-negligible impacts on the flow-induced vibration features.
The wake flow behind a stepped circular cylinder, which consists of two circular cylinders with different diameters, was investigated to obtain the flow interaction in the wakes of two cylinders [13] [14]. The dynamics of an oscillating stepped cylinder have received less attention. On the other hand, the vortex-induced vibration of a tapered circular cylinder has been studied by several researchers [15] [16] [17]. Seyed-Aghazadeh et al. investigated the influence of the tapered ratio on the vortex-induced vibration of circular cylinders free to oscillate in the transverse direction. They found that the lock-in range of a circular cylinder with a small tapered ratio is sustained through largely reduced velocities [17].
The energy harvesting studies using piezoelectric materials and vortex-induced motions were reviewed by Sodano et al. [18] and Rostami and Armendei [19]. The flow-induced vibrational motion of columnar structures is a L. O. A. Barata et al. Journal of Flow Control, Measurement & Visualization potential resource for the energy harvesting. The authors focused on the low-speed galloping vibration of the cantilever mounted rectangular prisms with a side ratio of D/H ≤ 0.5 to develop a vibrational power generation system using the flow-induced vibration and an iron-gallium alloy [12] [20].
The objectives of the present study are to improve the increment and stability of the response amplitude and the decrement of the vibration's starting velocity by modifying the prism shape. The effects of length (aspect ratio) and the size of the additional structure (a plate and a fin) attached to the surface of a rectangular prism on the transverse vibration characteristics of cantilevered rectangular prisms with a side ratio of D/H ≤ 0.5 have been investigated experimentally in a water tunnel. Figure 1 shows a schematic diagram of the test section of the water tunnel and measurement instruments. The experimental setup was similar to that as in [12] and hence will only be briefly described here. The experiment was performed in a water tunnel equipped with underground water. The rectangular test section of the water tunnel had a height of 400 mm, a width of 167 mm, and a length of 780 mm. The prism was mounted elastically to a plate spring attached to the ceiling wall of the test section with a jig. Figure 2 shows the test models of rectangular prisms. Table 1

Experimental Method
The uniform flow velocity U was varied from 0.74 to 2.7 m/s by controlling pump rotation speed and was measured using a pitot tube and digital differential pressure gauge (Nagano Keiki, GC50). The Reynold's number Re (=UH/v, where v is the kinematic viscosity of water) range was 1.4 × 10 4 to 5.4 × 10 5 . The reduced velocity V r (=U/f c H) was calculated from the characteristic frequency of the prism f c . Tip displacement y was measured using an acceleration sensor by using a personal computer. In order to prevent breakage of the plate spring, the maximum value of the prism amplitude was limited to η rms ≈ 0.2. The reduced mass damping Cn (=2mδ/ρDH, where m, δ, and ρ are the mass per unit length of the system, the logarithmic decrement of the structural damping parameter of a prism, and the water density, respectively) was measured by considering the initial displacement obtained by hitting the prism with a hammer in stationary water. The displacement amplitude η rms as converted into the angle of amplitude θ rms is shown in Figure 8. After the vibration begins, the angle of amplitude θ rms also increases linearly with the reduced velocity V r . The curves of the amplitude angle of the prisms with aspect ratios of L/H = 7.5 and 5 are in good agreement with those that have a large aspect ratio of L/H = 10. Although the displacement amplitude of the prism tip does not have the same displacement y rms as shown in Figure 6, the angle of amplitude θ rms is similar to that of the rectangular prisms with aspect ratios of L/H ≥ 5. However, the amplitude angle of the prisms with aspect ratio of L/H = 2.5 is smaller than that of those with an aspect ratio of L/H ≥ 5. The distinct difference in the vibration characteristic appears on a prism with a small aspect ratio of 2.5. It has a connection with the flow structure behind a finite prism with a critical aspect ratio, that is, L/H ≥ 5 [5]. Figure 9 and Figure     The time lapses of the tip displacement y for a rectangular prism of D/H = 0.5 with aspect ratios of L/H = 2.5 and 10 are shown in Figure 11 and Figure 12. In the case of a rectangular prism with D/H = 0.5, the amplitude of the vibration for a large aspect ratio of L/H = 10 did not even out, and its stability is the same as that for a small aspect ratio of L/H = 2.5. It seems that the unstable waveform is generated by the irregular flow reattachment on the side wall of the rectangular prism due to the vibration [1] [22].

Effect of Aspect Ratios of the Rectangular Prism
To evaluate the stability of a vibration, the variation in the nondimensional standard deviation of the peak displacement for a rectangular prism p θrms /θ rms with respect to the reduced velocity V r is shown in Figure 13. Here, p θrms is the root-mean-square (RMS) of the value of subtracting p θave from the absolute value of the peak displacement p θi for the waveform, and p θave is the time average of |p θi |. These values are given by the following equations: where N is the number of waveform peaks [12].   increase relatively more than that for the other prisms.

Effect of Added Plate Length
The effect of the configurations of additional structures, that is, a plate as shown in Figure 3 Figure 14 shows the nondimensional displacement amplitude η rms of a cantilevered rectangular prism with a side ratio of D/H = 0.2 and an aspect ratio of L/H = 10 for different lengths of plates. In the case of the added plate on the front surface of a rectangular prism with l/H = 2.5 (as shown in Figure 14 As shown in Figure 14(b), in the case of the added plate on the back surface of a rectangular prism, the characteristics were similar to the case of the added plate on the front surface of a rectangular prism. Therefore, if the length of the added plate was increased, the response amplitude of a cantilevered rectangular prism with D/H = 0.2 only shift to that of a cantilevered rectangular prism with D/H = 0.5. The rectangular prism with an added plate does not have a hybrid response amplitude, that is, the characteristics for a large response amplitude to the largely reduced velocity. Figure 15 and Figure 16 show the time lapses for tip displacement y and the nondimensional standard deviation of peak displacement p yrms /y rms for a rectangular prism of D/H = 0.2 with an added plate. The stability of the vibration of a rectangular prism with an added plate is not good. Because the vortex structure shedding from the rectangular prism with an added plate depends on the side ratio [1] [22], it changes along the length of the rectangular prism like the stepped circular cylinder [13]. Therefore, the p yrms /y rms of a rectangular prism of D/H = 0.2 with an added plate was larger than that of D/H = 0.    Figure 17 shows variation of second moment of area I 1 with the reduced velocities at the 15% non-dimensional response amplitude of a stepped rectangular prism Vr 0.15 . The second moment area I 1 (=I z ) of the stepped rectangular prism with respect to z-axis is given in Equation (2).
The second moment of area I 1 , that is the rigidity of prism, increased with increasing length of additional plate. Although the rigidity of prism increased by the plate-type additional structure, the reduced velocity at the 15% non-dimensional response amplitude of a stepped rectangular prism Vr 0.15 did not decrease. The plate-type additional structure does not give a good effect on the vibration characteristics, that is stable amplitude and decrease in velocity of onset vibration. The effect of fin-type additional structure on the response amplitude is described in the following section.

Effect of Added Fin Size
The effect of the configuration of a fin, which was fitted to the back of a rectan-   Figure 12. Figure 20 shows the variation of the nondimensional standard deviation of peak displacement of a rectangular prism p yms /y rms with respect to the reduced velocity Vr. The values of p yrms /y rms of the rectangular prism with a fin are larger than those of the rectangular prism without a fin. The variation with time of the tip displacement of a rectangular prism of D/H = 0.2 with a fin, shown in Figure 19(b), is similar to that of a rectangular prism of D/H = 0.5 without a fin, shown in Figure 12(b).
The stable flow-induced vibration was prevented by the fin and increased the stiffness of the thin rectangular prism.
Therefore, the relationship between the response amplitude and the flexural rigidity of the rectangular prism with a fin was examined. The second moment of area indicates the flexural rigidity of the rectangular prism. The second moment of area of the rectangular prismI 2 with respect to y-axis with a fin is given in Equation (3). shown in Figure 21. The second moments of cross-section area had similar values, that is, I 2 = 360 to 396 mm 4 . The curve of the response amplitude of the rectangular prism with a fin was in good agreement with that of the rectangular prism without a fin. Therefore, the reduced velocities at the 15% nondimensional response amplitude of a rectangular prism, Vr 0.15 , were plotted with respect to the second moment of cross-section area I of rectangular prisms with D/H= 0.1 to 0.5 in Figure 21 [12]. The experimental results indicate that the response amplitude of a rectangular prism depends on the second moment area of a rectangular prism. Further research on the wake structure of rectangular prism would clarify the relationship between the free-vibration characteristics, the fin shape, and the flexural rigidity of the rectangular prism.

Conclusions
The experiment on vibration characteristics of rectangular prisms with different aspect ratios and additional structures has been investigated in a water tunnel.
The main conclusions of the present study are as follows: 1) The reduced velocity of the vibration onset and the increment rate of the Journal of Flow Control, Measurement & Visualization 2) In the case of a rectangular prism with a plate, the reduced velocity of the initial vibration increases as the length of a plate increases. The values of the nondimensional standard deviation of the peak displacement of a rectangular prism with a plate are larger than those of a rectangular prism without a plate.
3) In the case of a rectangular prism with a fin, the reduced velocity of the initial vibration increases as the depth ratio of a fin increases. The values of the nondimensional standard deviation of the peak displacement of a rectangular prism with a fin are larger than those of a rectangular prism without a fin. The response amplitude of a rectangular prism depends on the second moment of area of the rectangular prism, independent of whether the prism has a fin.