Supercritical and Subcritical Hopf Bifurcations in a Delay Differential Equation Model of a Heat-Exchanger Tube Under Cross-Flow

Abstract

Nonlinear vibrations of a heat-exchanger tube modeled as a simply supported Euler–Bernoulli beam under axial load and cross-flow have been studied. The compressive axial loads are a consequence of thermal expansion, and tensile axial loads can be induced by design (prestress). The fluid forces are represented using an added mass, damping, and a time-delayed displacement term. Due to the presence of the time-delayed term, the equation governing the dynamics of the tube becomes a partial delay differential equation (PDDE). Using the modal-expansion procedure, the PDDE is converted into a nonlinear delay differential equation (DDE). The fixed points (zero and buckled equilibria) of the nonlinear DDE are found, and their linear stability is analyzed. It is found that stability can be lost via either supercritical or subcritical Hopf bifurcation. Using Galerkin approximations, the characteristic roots (spectrum) of the DDE are found and reported in the parametric space of fluid velocity and axial load. Furthermore, the stability chart obtained from the Galerkin approximations is compared with the critical curves obtained from analytical calculations. Next, the method of multiple scales (MMS) is used to derive the normal-form equations near the supercritical and subcritical Hopf bifurcation points for both zero and buckled equilibrium configurations. The steady-state amplitude response equation, obtained from the MMS, at Hopf bifurcation points is compared with the numerical solution. The coexistence of multiple limit cycles in the parametric space is found, and has implications in the fatigue life calculations of the heat-exchanger tubes.