Page 441 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 413

(ii) a ‘flow equation’, similar to those given in Section 5.2.8(b,c), namely

dt

(5.161)

where j = 1 for laminar and j = 2 for turbulent flow. The left-hand side of this equation
contains the inertial and frictional terms, while the right-hand side gives the vibratory
forcing due to lateral pipe motion; however, unlike in Bajaj et al. (1980) and Rousselet
& Henmann (1981), there is no upstream pressurization, so that fluid motion can be
induced only because of the mechanical vibration of the pipe.
The system is discretized and then integrated numerically. Appropriate analytical solu-
tions are obtained by the multiple-scales method for near-resonance conditions, i.e. when
the forcing frequency is close to one of the natural frequencies of the system.
It is found that, by means of nonlinear interaction, energy is transferred from the
vibration exciter to the fluid, resulting in nonzero mean flow velocity from the fixed
towards the free end, as well as a small oscillatory component. Typical results are shown
in Figure 5.71 for resonant excitation in the first and second modes of two different
pipes, showing that a substantial flow may be generated. These results are compared with
experiment in the figure, and agreement is exceptionally good.
Fluid flow damps the oscillation of the pipe. This effect is largest for the fundamental
mode, due to larger energy transfer to the fluid. Thus, the efficiency, measured as the
ratio of the kinetic energy imparted to the fluid compared to the energy supplied by the
shaker, decreases as the mode number increases.
Obvious uses of this discovery are for fluid transport or pumping, as well as transport of
granular materials - especially in cases where the fact that there are no internal moving
parts is important, e.g. in medicine or for corrosive or highly toxic substances.

5.1 1 CONCLUDING REMARKS

Further work on various aspects of the nonlinear dynamics of the system has been and
continues to be done, an attribute of this being a model system, of interest not only for
its own sake but also for developing theory or exemplifying dynamical behaviour in the
broad classes of nonconservative gyroscopic systems and fluidelastic systems.
Thus, in addition to linear studies on control of oscillations in pipes cited in Section 4.8,
Yau et a1 . (1 995) devise a successful and sophisticated control system for suppressing
chaotic oscillation of the constrained system of Figure 5.30, by means of so-called quan-
titative feedback theory (QFT).
Yoshizawa et al. (1997) study theoretically and experimentally the effect of lateral
harmonic excitation of a cantilevered pipe with an end-mass performing circular motion.
The state-of-the-art experimental set-up involves a laser displacement meter coupled to
an FFT analyser, two CCD video cameras and an on-line computer. Both theory and
experiments demonstrate a form of quenching. This phenomenon occurs when a self-
excited system performing limit-cycle oscillations is simultaneously subjected to forced
excitation; for sufficiently high amplitude of forcing, the character of the damping changes
completely (from ‘normally’ negative to positive) and the self-excited (free) oscillation is