The phenomenon of fluid-elastic instability and the velocity at which a heat exchanger tube bundle becomes unstable, known as the critical velocity, was discovered and empirically determined based upon single-span, linearly supported tube bundles. In this idealized configuration, the normal modes are well separated in frequency with negligible cross-modal contribution to the critical velocity. As a result, a critical velocity can be defined and determined for each mode. In an industrial heat exchanger or steam generator, not only do the tube bundles have multiple spans, they are also supported in over-sized holes. The normal modes of a multi-span tube bundle are closely spaced in frequency, and the non-linear effect of the tube-support plate interaction further promotes cross-modal contribution to the tube responses. The net effect of cross-modal participation in the tube vibration is to delay the instability threshold. Tube bundles in industrial exchangers often have critical velocities far above what were determined in the laboratory based upon single-span, linearly supported tube bundles. In this paper, the authors attempt to solve this non-linear problem in the time domain, using a time history modal superposition method. Time history forcing functions are first obtained by inverse Fourier transform of the power spectral density function used in classical turbulence-induced vibration analyses. The fluid-structure coupling force, which is dependent on the cross-flow velocity, is linearly superimposed onto the turbulence forcing function. The tube responses are then computed by direct integration in the time domain. By gradually increasing the cross-flow velocity, a threshold value is obtained at which the tube response just starts to diverge. The value of the cross-flow velocity at which the tube response starts to diverge is defined as the critical velocity of this non-linearly supported, multi-span tube bundle.
Discussion: “A Family of Hodograph Models for the Cross Flow Velocity Component of Three-Dimensional Turbulent Boundary Layers” (Shanebrook, J. R., and Hatch, D. E., 1972, ASME J. Basic Eng., 94, pp. 321–329)
