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.