A note on damping in heat-exchanger tubes subjected to cross-flow

  • Ajinkya Desai
  • Varun Vourganti
  • C. P. VyasarayaniEmail author


The equation governing the dynamics of a heat-exchanger tube is a delay differential equation (DDE). In all the earlier studies, only the stability boundaries in the parametric space of mass-damping parameter and reduced flow-velocity were reported. The contour plots showing the damping in different regions of the stability chart has never been reported, due to the complexity in solving the infinite-dimensional nonlinear eigenvalue problem associated with characteristic roots of the governing DDE. In this work using Galerkin approximations, the spectrum (characteristic roots) of the DDE is obtained. The rightmost characteristic root, whose real part represents the damping in the heat-exchanger tube is included in the stability chart. Interestingly, it is found that the highest damping is present in localized areas of the stability charts, which are close to the stability boundaries. These stability charts can be used to determine the optimal cross-flow velocities for operating the heat-exchanger tube for achieving maximum damping. Explicit evaluation of the characteristic roots allows us to show that the roots cross the stability boundary with a non-zero velocity, clearly indicating the existence of Hopf bifurcation at the stability boundary.


Heat-exchanger tube Delay differential equation Nonlinear eigenvalue problem Stability chart Damping contours 

List of symbols


Distance between centres of two adjacent cylinders of a row

\(\bar{\beta }\)

Fluid damping term for cylinder motion in lift direction

\(\bar{\kappa }\)

Fluid stiffness term for cylinder motion in lift direction

\(\bar{\lambda }\)

Dimensionless eigenvalue of a matrix


Mass-parameter of a tube \(\left( \frac{m}{\rho d^{2}}\right) \)

\(\delta \)

Modal logarithmic decrement

\(\lambda \)

Dimensionless roots of characteristic equation

\(\mathcal {O}\)

Order of magnitude

\(\mu \)

Flow retardation parameter

\(\omega _{n}\)

Natural angular frequency of tube

\(\rho \)

Fluid density

\(\tau \)

Dimensionless time-delay due to flow retardation \(\left( \frac{2\pi \mu }{aU_{r}}\right) \)


Modal viscous damping coefficient for cylindrical tube


Drag coefficient for stationary cylinder


Lift coefficient for flexible cylinder


Tube diameter


Modal stiffness coefficient for cylindrical tube


Natural frequency of tube \(\left( \frac{\omega _{n}}{2\pi }\right) \)


Streamwise distance between consecutive cylinder rows


Length of cylindrical tube


Mass per unit length of cylindrical tube


Pitch \((2T-d)\)


Dimensionless time

\(U_{\infty }\)

Free-stream velocity


Free-stream velocity at onset of transverse instability, also called critical velocity


Velocity of flow downstream of first row of upstream cylinders, also called gap velocity


Reduced critical velocity \(\left( \frac{U_{c}}{fd}\right) \)


Reduced flow-velocity \(\left( \frac{U_{\infty }}{fd}\right) \)


Nondimensional transverse displacement of flexible tube



The authors gratefully acknowledge the anonymous reviewers for reviewing this paper.


Funding was provided to C.P. Vyasarayani by the Department of Science and Technology through the Inspire fellowship (Grant Number DST/INSPIRE/04/2014/000972). The funders had no role in designing the study, collecting or analyzing data, the decision to publish, or preparing the manuscript

Compliance with ethical standards

Conflict of interest

The authors have no competing interests to declare.


  1. 1.
    Paidoussis MP, Price SJ, de Langre E (2011) Fluid-structure interactions: cross-flow-induced instabilities. Cambridge University Press, New York, pp 215–287zbMATHGoogle Scholar
  2. 2.
    Price SJ, Paidoussis MP (1984) An improved mathematical model for the stability of cylinder rows subject to cross-flow. J Sound Vib 97(4):615–640CrossRefGoogle Scholar
  3. 3.
    Granger S, Paidoussis MP (1996) An improvement to the quasi-steady model with application to cross-flow-induced vibration of tube arrays. J Fluid Mech 320:163–184CrossRefGoogle Scholar
  4. 4.
    Chen SS (1983) Instability mechanisms and stability criteria of a group of circular cylinders subjected to cross-flow. Part I: theory. J Vib Acoust Stress Reliab Des 105(1):51–58CrossRefGoogle Scholar
  5. 5.
    Khalifa A, Weaver D, Ziada S (2012) A single flexible tube in a rigid array as a model for fluidelastic instability in tube bundles. J Fluids Struct 34:14–32CrossRefGoogle Scholar
  6. 6.
    Hassan MA, Rogers RJ, Gerber AG (2011) Damping-controlled fluidelastic instability forces in multi-span tubes with loose supports. Nucl Eng Des 241(8):2666–2673CrossRefGoogle Scholar
  7. 7.
    Shinde V, Longatte E, Baj F, Braza M (2018) A theoretical model of fluidelastic instability in tube arrays. Nucl Eng Des 337:406–418CrossRefGoogle Scholar
  8. 8.
    Mahon J, Meskell C (2013) Estimation of the time delay associated with damping controlled fluidelastic instability in a normal triangular tube array. J Press Vessel Technol 135(3):030903CrossRefGoogle Scholar
  9. 9.
    Bazilevs Y, Takizawa K, Tezduyar T (2013) Computational fluid-structure interaction: methods and applications. Wiley, West Sussex, pp 9–19CrossRefGoogle Scholar
  10. 10.
    Khalifa A, Weaver D, Ziada S (2013) Modeling of the phase lag causing fluidelastic instability in a parallel triangular tube array. J Fluids Struct 43:371–384CrossRefGoogle Scholar
  11. 11.
    de Pedro Palomar B, Meskell C (2018) Sensitivity of the damping controlled fluidelastic instability threshold to mass ratio, pitch ratio and reynolds number in normal triangular arrays. Nucl Eng Des 331:32–40CrossRefGoogle Scholar
  12. 12.
    de Pedro B, Parrondo J, Meskell C, Oro JF (2016) CFD modelling of the cross-flow through normal triangular tube arrays with one tube undergoing forced vibrations or fluidelastic instability. J Fluids Struct 64:67–86CrossRefGoogle Scholar
  13. 13.
    Jiang N-b, Chen B, Zang F-g, Zhang Y-x (2015) An unsteady model for fluidelastic instability in an array of flexible tubes in two-phase cross-flow. Nucl Eng Des 285:58–64CrossRefGoogle Scholar
  14. 14.
    Liu J, Huang C, Jiang N (2014) Unsteady model for transverse fluid elastic instability of heat exchange tube bundle. Math Prob Eng 2014:942508Google Scholar
  15. 15.
    Meskell C (2009) A new model for damping controlled fluidelastic instability in heat exchanger tube arrays. Proc Inst Mech Eng Part A J Power Energy 223(4):361–368CrossRefGoogle Scholar
  16. 16.
    Hassan M, Hossen A (2010) Time domain models for damping-controlled fluidelastic instability forces in tubes with loose supports. J Press Vessel Technol 132(4):041302CrossRefGoogle Scholar
  17. 17.
    Chu I-C, Chung HJ, Lee S (2011) Flow-induced vibration of nuclear steam generator U-tubes in two-phase flow. Nucl Eng Des 241(5):1508–1515CrossRefGoogle Scholar
  18. 18.
    Vyasarayani CP, Subhash S, Kalmár-Nagy T (2014) Spectral approximations for characteristic roots of delay differential equations. Int J Dyn Control 2(2):126–132CrossRefGoogle Scholar
  19. 19.
    Hartlen R (1974) Wind-tunnel determination of fluid-elastic-vibration thresholds for typical heat-exchanger tube pattens. Ontario Hydro Research DivisionGoogle Scholar
  20. 20.
    Heilker WJ, Vincent RQ (1981) Vibration in nuclear heat exchangers due to liquid and two-phase flow. ASME J Eng Power 103:358–365CrossRefGoogle Scholar
  21. 21.
    Pettigrew M, Sylvestre Y, Campagna A (1978) Vibration analysis of heat exchanger and steam-generator designs. Nucl Eng Des 48(1):97–115CrossRefGoogle Scholar
  22. 22.
    Weaver DS, Grover L (1978) Cross-flow induced vibrations in a tube bank-turbulent buffeting and fluidelastic instability. J Sound Vib 59(2):277–294CrossRefGoogle Scholar
  23. 23.
    Weaver DS, El-Kashlan M (1981) The effect of damping and mass ratio on the stability of a tube bank. J Sound Vib 76(2):283–294CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Ajinkya Desai
    • 1
  • Varun Vourganti
    • 1
  • C. P. Vyasarayani
    • 1
    Email author
  1. 1.Department of Mechanical and Aerospace EngineeringIndian Institute of Technology HyderabadKandiIndia

Personalised recommendations