本文是以數值計算及實驗的方法探討橫流通過彈性排列圓管之振動現象。研究參數包括流速大小、圓管數目和自然振動頻率。在這些不同參數下，計算圓管的受力和位移，並與實驗結果比較。實驗中，圓管的振動以加速規量測，分析圓管的振幅、頻率與運動軌跡。決定圓管發生劇烈運動的臨界流速。研究結果顯示，當流速增加至臨界流速時，某些圓管會呈現劇烈振動，且有遲滯現象產生。在七根圓管的管列中，以第四列的圓管振動最劇。當流速增加至臨界流速以上時，圓管的響應頻率愈單一，其運動軌跡愈規則，尤其當圓管的自然振動頻率相當或接近鄰近圓管的振動頻率時。結果也顯示臨界流速的大小隨著上游相鄰圓管數目的增加而減少。另一方面，增加下游相鄰圓管對臨界流速的大小沒有顯著的影響。當圓管的自然振動頻率遠離鄰近圓管的振動頻率時，相鄰圓管的數目對臨界流速的大小沒有明顯影響。

ABSTRACT

The flow-induced vibrations of tubes in a rotated triangular array subject to cross flow are investigated numerically and experimentally. The parameters are inlet velocity of cross flow, number of tube, and tube natural frequency. In the study, the instantaneous fluid forces on tube surfaces are computed numerically, the instantaneous displacement of the tubes due to the fluid forces is calculated, and thus the motions of the tubes in cross flow are described. Experiments are also conducted to compare the numerical results. The tube vibrations in a water tunnel are measured by two accelerometers. The amplitudes, spectra, and trace of tube motion are presented. The critical velocities of tube vibrations are then determined. Experimental results show that some tubes vibrate seriously when the flow velocity increases up to a critical value, and hysteresis of the tube vibrations is observed. In case of the seven-tube array, the tubes in the fourth row exhibit the most serious vibration. When the flow velocity is above the critical value, only one dominant frequency of the tube vibrations is detected, comparing to multiple dominant frequencies in subcritical condition. Furthermore, the tube in supercritical condition behaves like a limit cycle, especially when the natural frequency is equal to or near the vortex shedding frequency from the upstream tubes. It is also shown that the critical velocity decreases with more surrounding tubes in the upstream and does not change as more adjacent tubes are added in the downstream. However, the tube number seems to have no effect on the critical velocity when the tube natural frequency is far from the vortex shedding frequency.

CONTENTS
Page
ABSTRACT ………………………………………………………………………… i
ACKNOWLEDGEMENTS………………………………………………………… ii
CONTENTS …………………………………………………………………………iii
LIST OF TABLES …………………………………………………………………. v
FIGURE CAPTIONS ………………………………………………………………vi
NOMENCLATURE ……………………………………………………………….xiv

CHAPTER 1. INTRODUCTION ………………………………………………… 1
1.1 Background ……………………………………………………………….. 1
1.2 Literature Review …………………………………………………………. 3
1.3 Motivations and Objectives ………………………………………………. 7
CHAPTER 2. COMPUTATIONAL FORMULATION …………………….……10
2.1 Calculation of the Flow Field.……………………………………………..10
2.2 Calculation of the Tube Motion.…………………………………………..13
CHAPTER 3. COMPUTAIONAL RESULTS AND DISCUSSION …………...15
3.1 Computational Domain and Conditions …………………………………..15
3.2 Test Calculations ………………………………………………………….16
3.3 Influence of Tube Number on Vibration of the Tubes in Cross Flow …….18
3.3.1 Vibration Amplitude ………………………………………….……..18
3.3.2 Orbits of Tube Vibrations.…………………………………………...23
3.3.3 Spectrum of Tube Vibration ………………………………………...26
3.4 Critical Flow Velocity …………………………………………………….27
3.5 Summary ………………………………………………………………….29
CHAPTER 4. EXPERIMENTAL ASPECTS ……………………………………31
4.1 Facility and Data Processing ……………………………………………..31
4.4.1 Water Tunnel and Instrumentation ………………………………… 31
4.4.2 Data Reduction and Uncertainty Analysis ………………………… 33
4.2 Influence of Tube Number on Tube Vibrations in Cross Flow ………….. 35
4.2.1 Vibration Amplitude ……………………………………………….. 35
4.2.2 Hysteresis of Tube Vibration ………………………………………..37
4.2.3 Orbit of Tube Vibration ……………………………………………. 39
4.2.4 Spectrum of Tube Vibration ……………………………………….. 41
4.3 Influence of Tube Natural Frequency on Vibration Amplitude of Tube Bundles in Cross Flow ……………………………………………………44
4.4 Critical Flow Velocity …………………………………………………….45
4.5 Summary ………………………………………………………………….47
CHAPTER 5. CONCLUSIONS …………………………………………………..49
REFERENCES …………………………………………………………………….51

REFERENCES

Blevins, R. D., Flow-Induced Vibration, Van Nostrand Reinhold, New York, NY (1990).

Cantwell, B. and Coles, D., “An Experimental Study of Entrainment and Transport in the Turbulent near Wake of a Circular Cylinder,” Journal of Fluid Mech., 136, pp. 321-374 (1983).

Chen, S. S., “Instability Mechanisms and Stability Criteria of a Group of Circular Cylinders Subjected to Cross Flow: Part I, Theory,” Journal of Vibration, Acoustic, Stress and Reliability in Design, 105, pp. 51-58 (1983).

Chen, S. S., “Guidelines for the Instability Flow Velocity of Tube Arrays in Cross-Flow,” Journal of Sound and Vibration, 93(3), pp. 439-455 (1984).

Chen, S. S., Zhu, S., and Cai, Y., “Experimental of Chaotic Vibration of Loosely Supported Tube Rows in Cross-Flow,” Transactions of the ASME, Journal of Pressure Vessel Technology, 117, pp. 204-212 (1995).

Chenoweth, J. M. and Kistler, R. S., “Tube vibrations in Shell-and-Tube Heat Exchangers,” HTRI Report prepared for ERDA (1976).

Connors, H. J., “Fluidelastic Vibration of Tube Arrays Excited by Cross Flow,” Paper presented at the Symposium on Flow Induced Vibration in Heat Exchangers, ASME Winter Annual Meeting, New York, (editor, Reiff, D. D.), pp. 42-56 (1970).

Connors, H. J., “Fluidelastic Vibration of Heat Exchanger Tube Arrays,” ASME Design Engineering Technical Conference, Chicago, Paper 77-DET-90, pp. 26-30 (1977).

Gere, James M. and Timoshenko, Stephen P., Mechanics of Materials, PWS-KENT, Boston, Massachusetts (1990).

Gorman, D. J., “Experimental Development of Design Criteria to Limit Liquid Cross-Flow-Induced Vibration in Nuclear Reactor Heat Exchange Equipment,” Nuclear Science and Engineering, 67, pp. 324-336 (1976).

Grover, L. K. and Weaver, D. S., “Cross-Flow Induced Vibrations in a Tube Bank-Vortex Shedding,” Journal of Sound and Vibration, 59, pp. 263-276 (1978).

Hartlen, R. T., “Recent Field Experience with Flow-Induced Vibration of Heat-Exchanger Tubes,” International Symposium on Vibration Problems in Industry, Keswick, England, Paper 611 (1973).

Ichioka, T., Kawata, Y., Nakamura, T., Izumi, H., Kobayashi, T., and Takamatsu, H., “Research on Fluid Elastic Vibration of Cylinder Arrays by Computational Fluid Dynamics (analysis of two cylinders and a cylinder row),” JSME International Journal, Series B, 40, pp.16-24 (1997).

Jendrzejczyk, J. A., Chen, S. S., and Wambsganss, M. W., “Dynamic Responses of a Pair of Circular Tubes Subjected to Liquid Cross Flow,” Journal of Sound and Vibration, 67(2), pp. 263-273 (1979).

Kassera, V., “Three Dimensional CFD-Analysis of Tube Vibrations Induced by Cross Flow,” 4th International Symposium on Fluid-Structure and Interaction, Aeroelasticity, Flow-Induced Vibration and Noise, ASME AD-Vol. 53-2, Volume II, pp. 137-143 (1997).

Kassera, V. and Strohmeier, K., “Experimental Determination of Tube Bundle Vibrations Induced by Cross-Flow,” Proceedings ASME Symposium on Flow-Induced Vibration, New York, (editor, Au-Yang, M. K.), PVP-273, pp. 91-97 (1994).

Kassera, V. and Strohmeier, K., “Simulation of Tube Bundle Vibrations Induced by Cross-Flow,” Journal of Fluids and Structures, 11, pp. 909-928 (1997).

Kolmogorov, A. N., “Equations of Turbulent Motion of an Incompressible Fluid,” Izvestia Academy of Science, USSR; Physics, 6, Nos. 1 and 2, pp. 56-58 (1942).

King, R. and John, D. J., “Wake Interaction Experiments with Two Flexible Circular Cylinders in Flowing Water,” Journal of Sound and Vibration, 45, pp. 259-283 (1976).

Kiya, M., Arie, M., Tamura, H., and Mori, H., “Vortex Shedding from Two Circular Cylinders in Staggered Arrangement,” Journal of Fluids Engineering, 102, pp. 166-173 (1980).

Lever, J. H. and Weaver, D. S., “On the Stability Behavior of Heat Exchanger Tube Bundles,” Parts I and II, 1984 Symp. on Flow Induced Vibrations, 2, ASME (1984).

Marn, J. and Catton, I., “Flow Induced Vibrations in Cylindrical Bundles: Two Dimensional Analysis Into Normal Modes,” Proc. 1991 National Heat Transfer Conference, HTD-165, Minneapolis, MN, pp. 9-14 (1991).

Marn, J., and Catton, I., “Analysis of Flow Induced Vibration Using the Vorticity Transport Equation,” Journal of Fluids Engineering, 115, pp. 485-492 (1993).

Moffat, R. J., “Describing the uncertainties in experimental results,” Experimental Thermal and Fluid Science, 1, pp.3-17 (1988).

Paidoussis, M. P., “Flow-Induced Vibrations in Nuclear Reactor and Heat Exchangers,” Proceedings of the IUTAM-IAHR Symposium on Practical Experiments with Flow-Induced Vibrations, Karlsruhe, Germany, 3-6 (1979).

Patankar, S. V., Numerical Heat Transfer and Fluid Flow, Hemisphere, New York (1980).

Pettigrew, M. J. and Gorman, D. J., “Vibration of Heat Exchanger Components in Liquid and Two-Phase Cross-Flow,” BNES International Conference on Vibration in Nuclear Plant, Keswick, U.K., Paper 2.3 (1978).

Prandtl, L., “Uber ein neues Formelsystem fur die ausgebildete Turbulenz,” Nacr. Akad. Wiss. Gottingen, Math-Phys. Kl., pp. 6-19 (1945).

Price, S. J., Paidoussis, M. P., and Giannias, N., “A Generalized Constrained Mode Analysis For Cylinder Arrays in Cross Flows,” 1988 Int. Symp. on Flow Induced Vibrations and Noise, 3, ASME (1988).

Sadaoka, N., Kobashi, K., and Umegaki, K., “Analysis of Flow-Induced Vibrations in Piping Systems and Circular Cylindrical Structures,” JSME International Journal, Series B, 41, pp. 221-226 (1998).

Schroder, K. and Gelbe, H., “Two- and Three- Dimensional CFD-Simulation of Flow-Induced Vibration Excitation in Tube Bundles,” Chemical Engineering and Processing, 38, pp. 621-629 (1999).

Shin, Y. S. and Wambsganss, W. N., “Flow-Induced Vibration in LMFBR Steam Generators,” ANL-75-16, A.N.L. Argonne, Illinois (1975).

Soper, B. M. H., “The Effect of Tube Layout on the Fluid Elastic Instability of Tube Bundles in Cross Flow,” ASME Symposium, HTD 9, Flow Induced Heat Exchanger Tube Vibration, Chicago, (editors Chenoweth, J. M. and Stenner, J. R.), pp. 1-9 (1980).

Soper, B. M. H., “The Effect of Tube Layout on the Fluid Elastic Instability of Tube Bundles in Cross Flow,” Journal of Heat Transfer, 105, pp. 744-750 (1983).

Tanaka, H., “Study on Fluidelastic Vibration of Tube Bundle,” Nihon Kikai Gakkai Ronbunshu (Japan Society of Mechanical Engineering), Section B, 46, pp. 1398-1407 (1980).

Tanaka, H., Takahara, S., and Ohta, K., “Flow-Induced Vibration of Tube Arrays with Various Pitch-to-Diameter Ratios,” Journal of Pressure Vessel Technology, 104, pp. 168-174 (1982).

Thompson, J. F., Warsi, Z. U. A., and Mastin, C. W., Numerical Grid Generation, North-Holland, Amsterdam (1985).

Weaver, D. S. and El-Kashlan, M., “On the Number of Tube Rows Required to Study Cross-Flow Induced Vibrations in Tube Banks,” Journal of Sound and Vibration, 75(2), pp. 265-273 (1981).

Weaver, D. S. and Grover, L. K., “Cross-Flow Induced Vibrations in a Tube Bank-Turbulent Buffeting and Fluid Elastic Instability,” Journal of Sound and Vibration, 59, pp. 277-294 (1978).

Weaver, D. S. and Yeung, H. C., “Approach Flow Direction Effects on the Cross-Flow Induced Vibrations of a Square Array of Tubes,” Journal of Sound and Vibration, 87(3), pp. 469-482 (1983).

Wilcox, D. C., Turbulence Modeling for CFD, CA:DCW Industries Inc., La Canada (1994).

Yetisir, M. and Weaver, D. S., “On an Unsteady Theory for Fluidelastic Instability of Heat Exchanger Tube Arrays,” 1988 Int. Symp. on Flow Induced Vibrations and Noise, 3, ASME (1988).

Yeung, H. C. and Weaver, D. S., “The Effect of Approach Flow Direction on the Flow-Induced Vibrations of a Triangular Tube Array,” American Society of Mechanical Engineers Journal of Vibration, Acoustics, Stress and Reliability in Design, 105, pp. 76-82 (1983).

Zdravkovich, M. M. and Namork, J. E., “Flow Structure within Both Stationary and Vibrating Tube Banks with Triangular Pitch,” Proceedings of the B.N.E.S. Symposium on Vibration in Nuclear Plant, Keswick, U.K., May 1978, Paper 2:5 (1978).

Zdravkovich, M. M. and Namork, J. E., “Structure of Interstitial Flow Between Closely Spaced Tubes in Staggered Array,” Proceedings ASME Symposium on Flow Induced Vibrations, (editors Chen S. S. and Bernstein M. D.), pp. 41-46 (1979).

Zhu, S., Cai, Y., and Chen, S. S., “Experimental Fluid-Force Coefficients for Wake-Induced Cylinder Vibration,” Journal of Engineering Mechanics, 121(9), pp. 1003-1015 (1995).

Zhu, S., Chen, S. S., and Cai, Y., “Vibration and Stability of Two Tubes in Cross-Flow,” Transactions of the ASME, Journal of Pressure Vessel Technology, 119, pp. 142-149 (1997).

Zukauskas, A., Ulinskas, R., Katinas, V., and Karni, J., Fluid Dynamics and Flow-Induced Vibrations of Tube Banks, Hemisphere, New York (1988).