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Trapped modes: an experimental investigation
Applied Ocean Research 23 (2001) 249±250
www.elsevier.com/locate/apor
Technical Note
Trapped modes: an experimental investigation C.H. Retzler* Department of Civil and Environmental Engineering, University of Southampton, High®eld, Southampton SO17 1BJ, UK Accepted 30 May 2001
1. Introduction The ¯uid around a vertical ®xed rigid circular cylinder in an open water channel can exhibit a local oscillation that does not propagate down the channel but is con®ned to the vicinity of the cylinder. Callan et al. [1] prove the existence of such a trapped mode for a cylinder of suf®ciently small radius, for ¯uid motion antisymmetric about the centreline of the channel and symmetric about a line through the centre of the cylinder perpendicular to the centreline. They also present a numerical solution for the frequency of the trapped mode as a function of cylinder diameter, not just for small diameters but for all diameters equal to or less than the channel width. For cylinder diameter 2a and channel width 2d, relevant quantities are the ratio a/d and frequency parameter kd, where k is the positive real root of v2 gk tanh kh; for water depth h and modal wave angular frequency v . Wave height is denoted by H.
2. Experimental arrangement A vertical surface-piercing circular cylinder was mounted centrally between the vertical parallel walls of a channel. The channel was 12.8 m long and 427 mm wide, with water depth 700 mm. The cylinder
* Fax: 144-23-8067-7519. E-mail address: [email protected] (C.H. Retzler).
bottom was mounted 167 mm above the channel ¯oor on a ball joint hinge and the cylinder top was connected to a linear actuator by a swivel-eye bearing. The cylinder was given a step impulse perpendicular to the channel centreline from an arbitrary rest position then held with its centre on the centreline. The resulting disturbance of the water surface was measured by 2 wave-gauges placed symmetrically either side of the cylinder on a line perpendicular to the channel centreline. Plastic foam beaches at both ends of the channel absorbed unwanted waves along the channel.
3. Data collection and processing Three cylinders were used in the tests, of diameters given in Table 1. Four equispaced measurements of the diameter of each cylinder were taken; the standard deviation (sd) is tabulated with the mean to indicate the relative roundness error. The cylinder diameters and channel width were measured to ^0.1 mm. Wavegauge data collection began 4 s after the end of the exciting impulse and continued for approximately 9 wave cycles. For each cylinder 16 identical tests were performed, with a 2 min wait between tests to allow the dissipation of waves. The wave height was about 15 mm, yielding a steepness parameter kH < 0:1: The ®rst antisymmetric mode accounted for virtually all the signal: total higher modes and noise were less than 5%, dropping to less than 3% when the 2 wavegauge records for each test were combined in antiphase. The period was measured from the linearly interpolated zero-crossing points of this combined record, for all the wave cycles in the record and for all 16 tests, yielding
0141-1187/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0141-118 7(01)00016-5
250
C.H. Retzler / Applied Ocean Research 23 (2001) 249±250
Table 1 Trapped modes: measurements and predictions Mean cylinder diameter (mm) ^ 1 sd
Mean a/d
Predicted kd a
Predicted frequency (Hz)
Measured frequency (Hz) ^ 1 sd
(Measured frequency ^ 1 sd)/ predicted frequency
159.8 ^ 1.1 200.5 ^ 0.0 323.5 ^ 1.3
0.3741 0.4696 0.7576
1.4619 1.4071 1.3183
1.3044 1.2797 1.2386
1.3009 ^ 0.0052 1.2779 ^ 0.0032 1.2338 ^ 0.0038
0.9973 ^ 0.0040 0.9986 ^ 0.0025 0.9961 ^ 0.0031
a
Numerical solution from Ref. [1].
the mean and standard deviation of the mode frequency. The wave height decay over the sample period was about 10%, implying a negligible reduction by damping of the natural frequency.
4. Results Results are summarized in Table 1. Each cylinder
diameter was associated with a ®rst antisymmetric mode of high purity and sharply de®ned frequency within 0.4% of the predicted trapped mode frequency.
References [1] Callan M, Linton CM, Evans DV. Trapped modes in twodimensional wave-guides. J Fluid Mech 1991;229:51±64.
www.elsevier.com/locate/apor
Technical Note
Trapped modes: an experimental investigation C.H. Retzler* Department of Civil and Environmental Engineering, University of Southampton, High®eld, Southampton SO17 1BJ, UK Accepted 30 May 2001
1. Introduction The ¯uid around a vertical ®xed rigid circular cylinder in an open water channel can exhibit a local oscillation that does not propagate down the channel but is con®ned to the vicinity of the cylinder. Callan et al. [1] prove the existence of such a trapped mode for a cylinder of suf®ciently small radius, for ¯uid motion antisymmetric about the centreline of the channel and symmetric about a line through the centre of the cylinder perpendicular to the centreline. They also present a numerical solution for the frequency of the trapped mode as a function of cylinder diameter, not just for small diameters but for all diameters equal to or less than the channel width. For cylinder diameter 2a and channel width 2d, relevant quantities are the ratio a/d and frequency parameter kd, where k is the positive real root of v2 gk tanh kh; for water depth h and modal wave angular frequency v . Wave height is denoted by H.
2. Experimental arrangement A vertical surface-piercing circular cylinder was mounted centrally between the vertical parallel walls of a channel. The channel was 12.8 m long and 427 mm wide, with water depth 700 mm. The cylinder
* Fax: 144-23-8067-7519. E-mail address: [email protected] (C.H. Retzler).
bottom was mounted 167 mm above the channel ¯oor on a ball joint hinge and the cylinder top was connected to a linear actuator by a swivel-eye bearing. The cylinder was given a step impulse perpendicular to the channel centreline from an arbitrary rest position then held with its centre on the centreline. The resulting disturbance of the water surface was measured by 2 wave-gauges placed symmetrically either side of the cylinder on a line perpendicular to the channel centreline. Plastic foam beaches at both ends of the channel absorbed unwanted waves along the channel.
3. Data collection and processing Three cylinders were used in the tests, of diameters given in Table 1. Four equispaced measurements of the diameter of each cylinder were taken; the standard deviation (sd) is tabulated with the mean to indicate the relative roundness error. The cylinder diameters and channel width were measured to ^0.1 mm. Wavegauge data collection began 4 s after the end of the exciting impulse and continued for approximately 9 wave cycles. For each cylinder 16 identical tests were performed, with a 2 min wait between tests to allow the dissipation of waves. The wave height was about 15 mm, yielding a steepness parameter kH < 0:1: The ®rst antisymmetric mode accounted for virtually all the signal: total higher modes and noise were less than 5%, dropping to less than 3% when the 2 wavegauge records for each test were combined in antiphase. The period was measured from the linearly interpolated zero-crossing points of this combined record, for all the wave cycles in the record and for all 16 tests, yielding
0141-1187/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0141-118 7(01)00016-5
250
C.H. Retzler / Applied Ocean Research 23 (2001) 249±250
Table 1 Trapped modes: measurements and predictions Mean cylinder diameter (mm) ^ 1 sd
Mean a/d
Predicted kd a
Predicted frequency (Hz)
Measured frequency (Hz) ^ 1 sd
(Measured frequency ^ 1 sd)/ predicted frequency
159.8 ^ 1.1 200.5 ^ 0.0 323.5 ^ 1.3
0.3741 0.4696 0.7576
1.4619 1.4071 1.3183
1.3044 1.2797 1.2386
1.3009 ^ 0.0052 1.2779 ^ 0.0032 1.2338 ^ 0.0038
0.9973 ^ 0.0040 0.9986 ^ 0.0025 0.9961 ^ 0.0031
a
Numerical solution from Ref. [1].
the mean and standard deviation of the mode frequency. The wave height decay over the sample period was about 10%, implying a negligible reduction by damping of the natural frequency.
4. Results Results are summarized in Table 1. Each cylinder
diameter was associated with a ®rst antisymmetric mode of high purity and sharply de®ned frequency within 0.4% of the predicted trapped mode frequency.
References [1] Callan M, Linton CM, Evans DV. Trapped modes in twodimensional wave-guides. J Fluid Mech 1991;229:51±64.