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Results from a series of studies on the stream-wise vibration of a circular cylinder verifying Japan Society of Mechanical Engineers Standard S012-1998,
*Guideline for Evaluation of Flow-induced Vibration of a Cylindrical Structure in a Pipe*, are summarized and discussed in this paper. Experiments were carried out in a water tunnel and in a wind tunnel using a two-dimensional cylinder model elastically supported at both ends of the cylinder and a cantilevered cylinder model with a finite span length that was elastically supported at one end. These cylinder models were allowed to vibrate with one degree of freedom in the stream-wise direction. In addition, we adopted a cantilevered cylinder model that vibrated with two degrees of freedom in both the stream-wise and cross-flow directions under the same vibration conditions as an actual thermocouple well. The value of the Scruton number (structural damping parameter) was changed over a wide range, so as to evaluate the value of the critical Scruton number that suppressed vibration of the cylinder. For the two-dimensional cylinder, two different types of stream-wise excitations appeared in the reduced velocity range of approximately half of the resonance-reduced velocity. For the stream-wise vibration in the first excitation region, due to a symmetric vortex flow, the response amplitudes were sensitive to the Scruton number, while the shedding frequency of alternating vortex flow was locked-in to half of the Strouhal number of vibrating frequency of a cylinder in the second excitation region. In addition, the effects of the aspect ratio of a cantilevered cylinder on the flow-induced vibration characteristics were clarified and compared with the results of a two-dimensional cylinder. When a cantilevered circular cylinder with a finite length vibrates with one degree of freedom in the stream-wise di-rection, it is found that acylinder with a small aspect ratio has a single excitation region, whereas a cylinder with a large aspect ratio has two excitation regions. Furthermore, the vibration mechanism of a symmetric vortex flow was investigated by installing a splitter plate in the wake to prevent shedding of alternating vortices. The vibration amplitude of acylinder with a splitter plate increased surprisingly more than the amplitude of a cylinder without a splitter plate. For a cantilevered cylinder vibrating with two degrees of freedom, the Lissajous figure of vibration of the first excitation region shows the trajectories of elongated elliptical shapes, and in the second excitation region, the Lissajous trajectories draw a figure “8”. The results and information from these experimental studies proved that Standard S012-1998 provides sufficient design methods for suppressing hazardous vibrations of cylinders in liquid flows.

Bluff bodies are structures having non-streamlined cross-sections, such as circular, square, and rectangular cylinders, around which flows separate from the body surface. The shear layers on surfaces of a bluff body build up, causing flow separation from the body, so as to form a region of reversed flow behind the body. Subsequently, vortex flows are periodically shed into the wake. The static fluid-dynamic characteristics of a stationary bluff body change greatly depending on the behavior of the shear flow and the wake [

When we scrutinized the results of the wind tunnel experiments by Scruton [

In this paper, we summarize and discuss the experimental results of circular cylinders of various shapes under different vibration conditions.

The water tunnel shown in

0.02. Reynolds number Re = Ud/ν, where ν is the kinematic viscosity of water, was in the range of 8 × 10^{3} to 4 × 10^{4}.

The two-dimensional circular cylinder model had a diameter d = 20 mm and effective span length L_{e} = 163 mm. Two circular end plates 80 mm in diameter (about 4d) and 0.5 mm in thickness were attached to both ends of the cylinder. _{e}/ρd^{2} (m is a mass unit span length of a cylinder in water), where δ is the logarithmic structural damping value of a cylinder vibrating in water. The response amplitude x of the cylinder vibrating in the stream-wise direction was detected by a laser displacement sensor, and the non-dimensional response amplitude ξ was defined as x/d. The fluctuating water velocity was detected by employing a hot-film probe located along the outer edge of the wake. In this way, we acquired the predominant values of the Strouhal number St_{w} = f_{w}d/U in the wake (where f_{w} is the fluctuating frequency of the wake flow).

_{e}/d, where L_{e} is span length) was varied from 5 to 21. The cylinder model was elastically supported by a leaf spring and rubber sheet attached to the ceiling of the test section. The natural frequency f_{c} of the cylinder in water was adjusted to be a constant value between 20 Hz to 30 Hz by using different leaf springs. The reduced velocity Vr (=U/f_{c}d) was varied from 0.9 to 4.0 by changing the water velocity. The logarithmic decrement of the structural

damping value of the vibrating cylinder model δ was varied by clamping aluminum plates and rubber plates with leaf-springs. The fluctuating displacement of the vibrating cylinder model was measured by an acceleration sensor mounted to the top of the model.

elastically supported by a total of four support springs at the base, that is, two springs in the stream-wise direction and two springs in the cross-flow direction. This arrangement allowed the model to vibrate in a rigid-body mode with two degrees of freedom. The structural damping value δ was easily changed by varying the distance between the upper and lower permanent magnets sandwiching a copper plate, and then the value of the Scruton number Cn was varied according to the structural damping value δ.

_{ms} of the cylinder in the stream-wise direction and the predominant values of Strouhal number St_{w} of fluctuating velocity in the wake, with respect to the reduced flow velocity Vr for a Scruton number of Cn = 0.80. From this figure, the vibration of the cylinder occurs around Vr = 1.4 and the amplitude ξ_{ms} linearly increases with respect to Vr, reaching ξ_{rms} = 0.068. In contrast, the lock-in phenomenon, where the frequency of a wake f_{w} equals f_{c}/4 of the natural frequency f_{c} for a vibrating cylinder, appears only in a narrow range around Vr = 1.5. This lock-in phenomenon is not observed in the wide range of Vr = 1.6 to 2.25 in the first excitation region. The Strouhal number of the wake St_{w} is almost constant and equals the Strouhal number for a stationary cylinder St_{n} of 0.18 to 0.19. Thus, the excitation

vibration of the cylinder was abruptly suppressed around Vr = 2.5, which is half of the reduced resonance velocity Vr_{cr} = U/f_{n}d = 1/St_{n}. The second excitation region appears in the high reduced-velocity region of Vr = 2.6 to 3.6, and the Strouhal number of fluctuating velocity in the wake St_{w} is locked to half of the Strouhal number of vibrating frequency of the cylinder (1/2St_{c}) in the range of Vr = 2.25 to 3.5. It is noted that this lock-in range includes the range near Vr = 2.25, where the damping force suppresses the vibration.

Next, _{rms} of the cylinder in the stream-wise direction against the reduced velocity Vr, when the Scruton number Cn was varied over the range 0.80 to 2.82 by employing an electromagnetic damper. At all values for Cn, the first and second excitation regions are present in the regions of 1.4 < Vr < 2.4 at less than Vr_{cr}/2 and the region of 2.6 < Vr < 3.5 at greater than Vr_{cr}/2, respectively. As the value of Cn increases, the amplitudes ξ_{rms} of the two excitation regions decrease, exhibiting different response characteristics with respect to Cn; that is, in the first excitation region, the maximum value of ξ_{rms} abruptly decreases according to the increase of the value of Cn, while the value of ξ_{rms} gradually decreases in the second excitation region, and the value of the on-set velocity of excitation increases. The behavior of the flow-induced vibration characteristics was clearly different with respect to Cn in the two excitation regions. _{cr}/2, as shown in

In prior experiments, Aguirre [

ξ_{rms} of the two-dimensional cylinder against the reduced velocity Vr, and compares it with the results for the cylinder without a splitter plate when the Scruton number Cn = 0.70. It was found that when a splitter plate was inserted, the excitation region extended from Vr = 2.0 up to 2.5, having a maximum amplitude of ξ_{ms} = 0.05 at Vr = 2.5. It is noteworthy that the response amplitudes ξ_{rms} of the cylinder both with and without a splitter plate were almost equal in the range of Vr = 1.1 to 2.2, where the symmetric vortices appeared. However, behind the cylinder without a splitter plate, alternating vortex flow formed and worked to suppress vibration in the range of Vr = 2.2 to 2.8. Additionally, the response characteristics of the lock-in region at St_{w} = 1/4St_{c} and l/2St_{c} for the vibrating cylinder with a splitter plate are different from those of the cylinder without a splitter plate, as shown in

_{max} of the vibrating two-dimensional circular cylinder against Scruton number Cn from the data obtained in a wind tunnel and a water tunnel [_{rms} values drops sharply in the first excitation region as the Cn values increase. Meanwhile, the decreasing rate of ξ_{max} against Cn value is relatively gentle in the second excitation region, and the value of ξ_{max}_{ }becomes as small as 0.005 at a Cn value of 2.5. At this point, the vibration is considered to be suppressed. Thus, we conclude from

vibration of a circular cylinder is suppressed under the defined conditions of Cn > 2.5 and Vr < 3.3. These results provide important data verifying JSME Standard S012-1998.

As shown in the visualization result of

characteristics of the cantilevered cylinders for which the aspect ratio AR was changed as AR = 5, 10, 14, and 21 and Cn was varied as Cn = 0.24 to 0.37 [_{rms} = 0.113 at the high velocity of Vr = 2.3, and then the amplitude gradually decreases between Vr = 2.5 and 3.2, resulting in a wide first excitation region accompanied by the symmetric vortex flow. When AR is smaller than 10, the second excitation region decays and vanishes due to the alternating vortex street, and only the first excitation region remains. However, when AR = 14 to 21, the vibration amplitudes are suppressed so as to form a valley at Vr = 2.4, and the amplitude increases again to the maximum value ξ_{max} = 0.082 in the vicinity of Vr = 2.7. Then, the second excitation region exists in the range of Vr = 2.5 to 3.2, and it has the same alternating vortex flow as the two-dimensional cylinder. It is clear that the characteristics of a cylinder with a finite span length greatly depend on the aspect ratio AR.

To clarify the excitation mechanism of stream-wise vibration, a splitter plate was inserted in a wake of a cantilevered cylinder with AR = 10 to observe its effect on vortex shedding.

maximum value of ξ_{max} = 0.12 at Vr = 2.8 and sharply decreases at Vr = 2.9 to 3.0. It is noteworthy that the two response curves are exactly the same in the region from Vr = 1.1 to 2.0, where the symmetric vortices are formed, and the alternating vortex flow suppresses the vibration of the cylinder without a splitter plate in the range from Vr = 2.2 to 3.0.

_{rms} in the stream-wise direction and η_{rms} in the cross-flow direction against the reduced velocity Vr as the experimental results of vibration of a cantilevered cylinder with a span length of AR = 20 (d = 0.015 m, L = 0.3 m). From this figure, it is clear that the cylinder vibration starts from Vr = 1.25 and reaches a peak at Vr = 2.3 in the first excitation region. As shown in _{cr}/2 = 2.4), and the amplitudes increase again in the higher reduced velocity range. In the second excitation region, the amplitude attains a maximum value at Vr = 3.0, and the Lissajous trajectory draws a figure “8” at Vr = 2.99 in

However, in the case of a short cylinder with the span length AR = 10, the two-dimensionality of alternating wake vortex flows is weakened by the flow veil over the cylinder tip, and only the first excitation region of the symmetric vortex flow still remains. The values of the wake Strouhal number St_{w} are locked to 1/4St_{c} of the Strouhal number of vibrating frequency of the cylinder St_{c} (=1/Vr_{cr}) at a lower velocity in the first excitation region. Then the value of St_{w} remains constant, approximately St_{w} = 0.2 around Vr = 2.5. Furthermore, the value of St_{w} is locked to 1/2St_{c} over the region above Vr = 2.25, including the suppression region and the second excitation region.

We analyzed the results of the stream-wise vibration of various circular cylinders and evaluated the critical values of the Scruton number Cn that is able to sufficiently suppress vibration. To compare the response characteristics of various cylinders with different aspect ratios, the maximum amplitudes ξ_{max} (=x_{max}/d, where x_{max} is the maximum RMS value of the response amplitude of a cylinder vibrating in the stream-wise direction) with respect to the Cn values are summarized in

ξ_{max} of the first and second excitation regions decrease but exhibit different response characteristics against the Cn values. That is, the maximum amplitude values ξ_{max} sharply decrease against the increase of Cn values in the first excitation region less than Vr_{cr}/2. Meanwhile, the rate of decrease of ξ_{max} against Cn is relatively gentle in the second excitation region, and the value of ξ_{max} becomes as small as 0.005 at Cn = 2.5, which implies that the vibration is suppressed.

The maximum amplitude values ξ_{max} of the cantilevered cylinders of AR = 10 and 21 vibrating with one degree of freedom are shown by symbols against various Cn values in both two excitation regions in _{max} values of the cantilevered cylinders (finite span) decrease more quickly with increasing Cn values, as compared with the curve of the two-dimensional cylinder.

Finally, _{max} values in the first excitation region are the same as those of the cantilevered cylinders with one degree of freedom in

Results from a series of studies on the stream-wise vibration of a circular cylinder that verified Japan Society of Mechanical Engineers Standard S012-1998 were summarized and discussed. Experiments were carried out with a two-dimensional cylinder model and a cantilevered cylinder model with a finite span length. These cylinder models were allowed to vibrate with one degree of freedom in the stream-wise direction. In addition, we employed a cantilevered cylinder model that vibrated with two degrees of freedom in both the stream-wise and cross-flow directions under the same vibration conditions as an actual thermocouple well. The Scruton number Cn was changed over a wide range, so as to evaluate the critical value. For the two-dimensional cylinder, two different types of stream-wise excitations appeared in the range of approximately half of the resonance-reduced velocity. For the stream-wise vibration in the first excitation region due to the symmetric vortex flow, the response amplitudes were sensitive to Cn, while the shedding frequency of alternating vortex flow was locked-in to half of the Strouhal number of vibrating frequency of a cylinder in the second excitation region. When a cantilevered circular cylinder with a finite length vibrates with one degree of freedom in the stream-wise direction, a cylinder with a small aspect ratio has a single excitation region, whereas a cylinder with a large aspect ratio has two excitation regions. Furthermore, the vibration mechanism due to the symmetric vortex flow was investigated by installing a splitter plate in the wake to prevent shedding of alternating vortices. The vibration amplitude of a cylinder with a splitter plate increased surprisingly more than the amplitude of a cylinder without a splitter plate. For a cantilevered cylinder vibrating with two degrees of freedom, the Lissajous figure of vibration of the first excitation region shows the trajectories of elongated elliptical shapes, and in the second excitation region, the Lissajous trajectories draw a figure “8”.

The results and information from these experimental studies demonstrated that Standard S012-1998 is sufficient as a general standard for evaluation of flow-induced vibration of circular cylinders applicable to structures in various industrial fields, including offshore structures, chimneys, and structures in chemical plants and power plants

The work described in this paper was mainly based on the research of the flow-induced vibration of bluff bodies that the authors have carried out with former students at Kanazawa University, including Dr. A. Nakamura, Mr. T. Kosugi, Prof. T. Yasuda, Mr. T. Iwasaki, Mr. H. Uchida, Mr. R. Tamaki, Mr. Y. Nagamori, and Mr. F. Matsunaga.

The authors thank Mr. T. Kuratani for his help at Kanazawa University, Prof. K. Matsuda, Dr. S. Saitoh, and Mr. T. Sugimoto (at that time, IHI Co.) for their technical assistance.

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

Okajima, A. and Kiwata, T. (2019) Flow-Induced Stream-Wise Vibration of Circular Cylinders. Journal of Flow Control, Measurement & Visualization, 7, 133-151. https://doi.org/10.4236/jfcmv.2019.73011

AR = aspect ratio of cylinder, L_{e}/d (-)

Cn = Scruton number (reduced structural mass-damping parameter) 2 mL_{e}δ/ρd^{2}, 2 ML_{e}δ (-)

d = diameter of cylinder (m)

f_{c} = natural frequency of vibrating cylinder in water (Hz)

f_{n} = natural vortex shedding frequency in wake for stationary cylinder (Hz)

f_{w} = vortex shedding frequency in wake (Hz)

l = length of splitter plate (m)

L = total spanlength of cylinder (m)

L_{e} = effective spanlength of cylinder (m)

m = mass per unit total spanlength in water (kg/m)

Re = Reynolds number, Ud/ν (-)

St_{c} = Strouhal number of vibrating cylinder, f_{c}d/U (-)

St_{n} = Strouhal number for stationary cylinder, f_{n}d/U (-)

St_{w} = Strouhal number of vortex shedding frequency in wake, f_{w}d/U (-)

U = uniform flow velocity (m/s)

Vr = reduced velocity, U/f_{c}d (-)

Vr_{cr} = reduced resonance velocity, U/f_{n} d = 1/St_{n} (-)

x_{max} = maximum RMS response amplitude of cylinder in stream-wise direction (m)

x_{rms} = root-mean-square (RMS) response amplitude of cylinder in stream-wise direction (m)

y_{rms} = RMS response amplitude of cylinder in cross-wise direction (m)

δ = logarithmic decrement of structural damping parameter in water (-)

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

ν = kinematic viscosity of working fluid (m^{2}/s)

ξ_{max} = non-dimensional maximum value of RMS response amplitude of vibrating cylinder in the stream-wise direction, x_{max}/d (-)

ξ_{rms} = non-dimensional value of RMS response amplitude of cylinder vibrating in stream-wise direction, x_{rms}/d (-)

ρ = fluid mass density (kg/m^{3})