^{1}

^{*}

^{2}

^{2}

^{3}

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.

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 [

The flow around a finite-length prism without an end plate becomes a strongly three-dimensional structure [

The flow-induced vibration of a cantilever rectangular prism has been investigated by a few researchers [

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 [

The energy harvesting studies using piezoelectric materials and vortex-induced motions were reviewed by Sodano et al. [

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.

L/H | Spring thickness (mm) | D/H | |||||
---|---|---|---|---|---|---|---|

0.2 | 0.5 | ||||||

f_{n} (Hz) | δ | C_{n} | f_{n} (Hz) | δ | C_{n} | ||

2.5 | 0.4 | 34.4 | 0.111 | 1.95 | 43.1 | 0.016 | 0.28 |

0.6 | 31.1 | 0.044 | 0.77 | 26.4 | 0.023 | 0.40 | |

5 | 0.4 | 23.8 | 0.035 | 0.61 | 15.1 | 0.036 | 0.63 |

0.6 | 38.7 | 0.026 | 0.46 | 24.8 | 0.026 | 0.46 | |

0.8 | - | - | - | 33.9 | 0.017 | 0.30 | |

7.5 | 0.6 | 26.4 | 0.061 | 1.08 | 16.5 | 0.030 | 0.52 |

0.8 | 36.4 | 0.025 | 0.44 | 22.7 | 0.023 | 0.40 | |

1.0 | 46.1 | 0.023 | 0.41 | 28.8 | 03017 | 0.30 | |

10 | 0.8 | 26.5 | 0.026 | 0.46 | 16.4 | 0.031 | 0.55 |

1.0 | 33.2 | 0.022 | 0.39 | 20.6 | 0.022 | 0.39 | |

1.2 | 38.9 | 0.020 | 0.35 | 24.3 | 0.025 | 0.44 |

mm and cross-section height of h = 20 mm to the front or back of the cantilevered rectangular prism with D/H = 0.2 by screws. The stepped rectangular prism with the additional plate of different lengths was investigated to determine whether the stepped rectangular prism had hybrid vibration characteristics of the D/H = 0.2 and 0.45 prisms. The length plate ratio l/H was varied from 2.5 to 7.5. _{c} of the prism with additional plate and fin increased from 0.3 to 3.7 Hz more than that without them.

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 (Showa Measuring Instrument, 2302CW) implanted inside the tip of the prism and an integrator (RION UV-12 and UV-05). The signal output of the integrator was converted using a 12-bit A/D converter with a sampling frequency of 2 kHz, and 10,000 data points were recorded. The characteristic frequency of test models f_{c} was measured using the FFT analyzer (ONO SOKKI, CF-5201). The characteristic frequency f_{c} of the prism was set in a constant value from 16.4 Hz to 38.9 Hz using different thicknesses of plate springs. The root-mean-square (RMS) values of the fluctuation of displacement of a prism tip y_{rms}, the non-dimensional displacement amplitude η_{rms} (=y_{rms}/H), and the angle amplitude θ_{rms} [≈tan^{−1{}y_{rms}/(L+ L’)}∙180/π, where L’ is length of a jig] were calculated 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.

_{rms} of a cantilevered rectangular prism with an aspect ratio of L/H = 10 for different natural frequencies using different thicknesses of plate springs. After the vibration begins, the amplitude η_{rms} increases linearly with the reduced velocity Vr. Although the velocity in this experiment is a different range of Reynolds number, each curve of the non-dimensional displacement amplitude η_{rms} has good agreement. The reduced velocity of the vibration onset and the non-dimensional displacement amplitudes do not depend on the test models’ characteristic frequency in still water, f_{c}. The onset vibration of rectangular prisms with D/H = 0.2 and 0.5 occurred near V_{r} ≈ 1.5 and 3.0, respectively. Thus, the reduced velocity of the vibration onset for prisms with the small side ratio of D/H = 0.2 is lower than that for the large side ratio of D/H = 0.5. The reduced resonance velocities for the rectangular prisms with side ratios of D/H = 0.5 and 0.2 were Vr_{cr} ≈ 8.0 and 7.3 [_{r}_{cr}.

The effect of aspect ratio on the nondimensional displacement amplitude η_{rms} and the ratio of the actual vibration frequency of the prism in a flow on the

characteristic frequency of the still water, f/f_{c}, is shown in _{rms} increases with an uptick in the reduced velocity V_{r}. The frequency ratio f/f_{c} decreases slightly with an increase in the reduced velocity V_{r}. It seems that the vibration frequency in a fluid depends on the vibration amplitude. Namely, it is affected by the added (virtual) mass and viscosity. _{rms}/dV_{r}_{ }of prisms from the onset of vibration to the end of measurement. The value of dη_{rms}/dV_{r} rises with an increase in the aspect ratio L/H. The increment rate of a rectangular prism with D/H = 0.2 is larger than that with D/H = 0.5 for 2.5 ≤ L/H < 10. For the rectangular prisms with D/H = 0.2, the increment rate keeps constant over L/H = 7.5.

The displacement amplitude η_{rms} as converted into the angle of amplitude θ_{rms} is shown in _{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 _{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 [

_{r} = 1.75 and 3.2. On the other hand, for L/H = 10, the amplitude of the vibration has a uniform amplitude for V_{r} = 2.0 and 3.15. These phenomena are causally related to the two-dimensionality of the

flow structure of the wake, that is, the interaction between the lengthwise vortices and a tip vortex, in which the Kármán vortex shedding is suppressed and replaced with arch vortex formation for the small aspect ratio [

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

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 _{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:

p θrms = ∑ i = 1 N ( | p θi | − p θave ) 2 N , p θave = ∑ i = 1 N | p θi | N (1)

where N is the number of waveform peaks [_{θrms}/θ_{rms} are large. For a largely reduced velocity, the p_{θrms}/θ_{rms} of the rectangular prisms with a small aspect ratio of L/H = 2.5 are larger than for those with a large aspect ratio. The stability of the vibration for a prism with a side ratio of D/H = 0.2 and an aspect ratio of L/H ≥ 5 tends to

increase relatively more than that for the other prisms.

The effect of the configurations of additional structures, that is, a plate as shown in

D/H = 0.2 and 0.45.

_{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

As shown in

_{y}_{rms}/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 [_{y}_{rms}/y_{rms} of a rectangular prism of D/H = 0.2 with an added plate was larger than that of D/H = 0.2 without an added plate at the same reduced velocity. The characteristics shifted to those of a rectangular prism with D/H = 0.5.

The relationship between the response amplitude and the flexural rigidity of the stepped rectangular prism was examined. _{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).

I 1 ( = I z ) = l ( D + d ) 3 + ( L − l ) D 3 3 − { l ( D + d ) + ( L − l ) D } e 1 3 e 1 = 0.5 l ( D + d ) 2 + ( L − l ) D 2 l ( D + d ) + ( L − l ) D (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.

The effect of the configuration of a fin, which was fitted to the back of a rectangular prism, on flow-induced vibration of a rectangular prism with D/H = 0.2 was investigated. 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 sizes of fins are shown in _{rms} increased linearly with an uptick in the reduced velocity V_{r}, which was similar to the response amplitude of the rectangular prism without a fin. The reduced velocity of the vibration onset went up with an increase in the depth ratio of the fin, d/H. The increment rate of the nondimensional response amplitude

η_{rms} for the reduced velocity Vr of a rectangular prism without a fin was higher than that of a prism with a fin. The increment rate dη_{rms}/dVr of the prism with a fin of d/H = 0.35 and 0.4 is low in the region of initial vibration.

_{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

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).

I 2 ( = I y ) = H D 3 + h d 3 3 + h d D ( D + d ) − { h d ( D + d 2 ) + H D 2 2 h d + H D } 2 ( H d + h d ) (3)

where the quantities are defined in the nomenclature section. For example, the values of the second moment of area of a rectangular prism with D/H = 0.2 with and without a fin (h/H, d/H) = (0.25, 0.15) were 273 mm^{4} and 107 mm^{4}, respectively. The flexural rigidity of a rectangular prism with a fin is 2.6 times that of a rectangular prism without a fin.

The response amplitude η_{rms} of the second moment of area of the rectangular

prisms with (D/H, h/H, d/H) = (0.3, 0, 0), (0.2, 0.5, 0.15), and (0.2, 0.25, 0.2) are shown in _{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

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

response amplitude of displacement are influenced by the aspect ratio of L/H. The vibration characteristics of angle amplitude of the prisms with aspect ratio L/H ≥ 5.0 are not similar to those with aspect ratios below the critical aspect of L/H = 2.5. The prisms with an aspect ratio of L/H = 2.5 have nonuniform amplitude, small angle amplitude, and a low increment rate of the response amplitude. The prisms with L/H = 10 have stable vibration and a high increment rate of the response amplitude.

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.

This work was supported by JST CREST Grant JPMJCR15Q1 (15665024). The authors are thankful to Professor Emeritus Dr. Okajima for productive discussions and to technician Mr. Kuratani and students Mr. Yamaguchi, Mr. Mizukami and Mr. Nagase for their help with the experiment.

The authors declare no conflicts of interest regarding the publication of this paper.

Barata, L.O.A., Kiwata, T., Kono, T. and Ueno, T. (2020) Effects of Span Length and Additional Structure on Flow-Induced Transverse Vibration Characteristic of a Cantilevered Rectangular Prism. Journal of Flow Control, Measurement & Visualization, 8, 102-120. https://doi.org/10.4236/jfcmv.2020.83006

C_{n} = reduced mass-damping parameter of the system, 2mδ/ρDH

D = depth of a prism in the flow direction

f = vibration frequency of the prism in a flow

f_{c} = characteristic frequency of the prism in a still water

f_{n} = natural frequency for vortex shedding from a fixed prism

H = height of a prism normal to the flow direction

L = span length of a prism

l = length of a plate

m = mass per unit length of the system

N = number of data

p_{ave} = average peak amplitude

p_{i} = peak amplitude

p_{rms} = root-mean-square of the peak amplitude

Re = Reynolds number, UH/ν

U = uniform flow velocity

u = x-component velocity

V_{r} = reduced velocity, U/f_{c}H

V_{r}_{cr} = reduced resonance velocity, U/f_{n}H

V_{r}_{0.15} = reduced velocity at the 15% non-dimensional response amplitude of a prism

y = displacement of the prism tip

y_{rms} = root mean square of displacement amplitude of the prism tip

δ = logarithmic decrement of the structural damping parameter of a prism

θ_{rms} = root mean square of amplitude angle of the prism tip

η_{rms} = non-dimensional value of RMS response amplitude of cylinder vibrating in cross-flow direction, y_{rms}/H(−)

ν = kinematic viscosity of water

ρ = water density