Effect of nonlinear cladding stiffness on the stability and Hopf bifurcation of a heat-exchanger tube subject to cross-flow


The linear stability of a heat-exchanger tube modeled as a single-span cantilever beam subjected to cross-flow has been studied with two parameters: (1) varying stiffness of the baffle-cladding at the free end and (2) varying flow velocity. A mathematical model incorporating the motion-dependent fluid forces acting on the beam is developed using the Euler–Bernoulli beam theory, under the inextensible condition. The partial delay differential equation governing the dynamics of the continuous system is discretized to a set of finite, nonlinear delay differential equations through a Galerkin method in which a single mode is considered. Unstable regions in the parametric space of dimensionless cladding stiffness and flow velocity are identified, along with the magnitude of damping in the stable region. This information can be used to determine the cladding stiffness at which the system should be operated to achieve maximum damping at a known operational flow velocity. Furthermore, the system is found to lose stability by Hopf bifurcation and the method of multiple scales is used to analyze its post-instability behavior. Stable and unstable limit cycles are observed for different values of the linear component of the dimensionless cladding stiffness. A global bifurcation analysis indicates that the number of limit cycles decreases with increasing linear cladding stiffness. An optimal range for the linear cladding stiffness is recommended where tube vibrations would either diminish to zero or assume a relatively low amplitude associated with a stable limit cycle.

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L :

Tube length (m)

\(\zeta\) :

Damping ratio

\(\rho\) :

Fluid density (kg m\(^{-3}\))

D :

Outer diameter of tube (m)

t :

Dimensionless time

X :

Axial spatial coordinate (m)

T :

Time (s)

\(\mu\) :

Flow retardation parameter

\({\mathcal {O}}\) :

Order of magnitude

x :

Dimensionless axial spatial coordinate

w :

Dimensionless transverse displacement of tube

\(\tilde{f}\delta (X-X_b)\) :

Restraining force due to spring at \(X=X_b\) (N m\(^{-1}\))

A :

Cross-sectional area of cylinder (\(\hbox {m}^2\))

W :

Transverse displacement of tube (m)

\(\tilde{U}\) :

Cross-flow velocity (m s\(^{-1}\))

\(f\delta (x-x_b)\) :

Dimensionless spring force

E :

Young’s modulus of tube material (Pa)

\(C_{D}\) :

Drag coefficient

\(C_{L}\) :

Lift coefficient

M :

Mass per unit length of tube (kg m\(^{-1}\))

C :

Viscous damping coefficient (N s m\(^{-2}\))

I :

Second moment of inertia of tube cross-section (m\(^{4}\))

\(\delta _1\) :

Modal logarithmic decrement

\(C_{ma}\) :

Added-mass coefficient

\(\varDelta T\) :

Time delay (s) \(\left( \frac{\mu D}{ \tilde{U}}\right)\)

m :

Tube mass-parameter \(\left( \frac{M}{\rho D^{2}}\right)\)

U :

Dimensionless flow-velocity \(\left( \frac{2\pi \tilde{U}}{D\varOmega _1}\right)\)

\(\omega _{cr}\) :

Dimensionless critical angular frequency

\(\tau\) :

Dimensionless time delay due to flow retardation \(\left( \frac{2\pi }{U}\right)\)

q :

Dimensionless tube vibration response

\(\varOmega _1\) :

Frequency of first mode (\(\hbox {s}^{-1}\))

\(\lambda _1\) :

Dimensionless eigenvalue of first mode

F :

Cross-flow-induced force per unit length (N m\(^{-1}\))

\(K_1\) :

Linear spring stiffness (N m\(^{-1}\))

\(K_2\) :

Cubic spring stiffness (N m\(^{-3}\))

\(k_1\) :

Dimensionless linear spring stiffness \(\left( \frac{K_1L^4}{\lambda _1^4EID}\right)\)

\(k_1^{cr}\) :

Dimensionless critical linear spring stiffness

\(k_2\) :

Dimensionless cubic spring stiffness \(\left( \frac{K_2L^6}{\lambda _1^4EID}\right)\)

P :

Pitch (m)


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The authors gratefully acknowledge the Department of Science and Technology for funding this research through the Inspire Fellowship (DST/INSPIRE/04/2014/000972).

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Correspondence to Ajinkya Desai.

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Vourganti, V., Desai, A., Samukham, S. et al. Effect of nonlinear cladding stiffness on the stability and Hopf bifurcation of a heat-exchanger tube subject to cross-flow. Meccanica (2020).

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  • Heat-exchanger tube
  • Linear stability
  • Nonlinear cladding stiffness
  • Delay differential equation
  • Method of multiple scales
  • Hopf bifurcation