Nearly six decades ago, Stoker, in the introduction to his seminal book on nonlinear vibrations (Stoker 1950), pointed out ‘It is perhaps worth while to consider for a moment the reasons why one should be interested particularly in the nonlinear problems in mechanics. Basically the reason is, of course, that all of the problems in mechanics are nonlinear from the outset, and the linearizations commonly practiced are an approximating device which is often a confession of defeat in the challenge presented by nonlinear problems as such’. This powerful statement is echoed in the celebrated book on nonlinear oscillators (Andronov et al. 1966), which provides a very comprehensive source of the theoretical foundations and applications in the area of nonlinear oscillations. The concepts of slow and fast oscillations, small and large terms in solutions, and periodicity have been clearly introduced in this book by solving various dynamical problems. In more recent works (e.g. Guckenheimer & Holmes 1983; Thompson & Stewart 1986), one can find a systematic approach to model and analyse nonlinear systems exhibiting chaos.
Since the invention of digital computers, the interest in the area of nonlinear mechanics, and nonlinear dynamics in particular, has grown nearly in an exponential manner. Currently we have in excess of 40 journals that publish nonlinear dynamics results. However, there is a huge disproportion between numbers of theoretical and experimental papers. A very conservative estimate would account for approximately 5% of experimental articles. Experiments are vital to confirm new theories, make viable predictions and ultimately to establish firm foundations for developments in science and technology.
The main aim of arranging and compiling this theme is to perhaps address in a small but powerful way the misbalance mentioned above. The theme comprising a total 12 papers has two issues ‘Experimental nonlinear dynamics I. Solids’ and ‘Experimental nonlinear dynamics II. Fluids’.
This second part of the theme is formed from five papers written by world experts in the areas of fluid–structure interactions and fluids. The first two articles by Benaroya & Gabbai (2008) and Modarres-Sadeghi et al. (2008) form a bridge between solids and fluids as they consider fluid–structure interactions. The following three contributions are focused on the nonlinear effects originating mainly from fluids.
In the first paper on ‘Modelling vortex-induced fluid–structure interaction’ Benaroya & Gabbai (2008) provide an extensive review with a focus on offshore structures. Their primary aim is to generalize the Hamilton's variational framework so that systems of flow oscillator equations can be derived from first principles. It is demonstrated here that flow oscillator models are a subclass of the general, physical-based framework. A flow oscillator system is a reduced-order mechanical model, generally comprising two mechanical oscillators, one modelling the structural oscillation and the other a nonlinear oscillator representing the fluid behaviour coupled to the structural motion. Reduced-order analytical model development continues to be carried out using a Hamilton's principle based variational approach. This provides flexibility in the long run for generalizing the modelling paradigm to complex, three-dimensional problems with multiple degrees of freedom, although such extension is very difficult. As both experimental and analytical capabilities advance, the critical research path in developing and implementing fluid–structure interaction models entails to (i) formulating generalized equations of motion, as a superset of the flow-oscillator models and (ii) developing experimentally derived, semi-analytical functions to describe key terms in the governing equations of motion.
Modarres-Sadeghi et al. (2008) discuss experiments on vertical slender flexible cylinders clamped at both ends and subjected to axial flow. Three series of experiments were conducted in order to observe the dynamical behaviour of the system, and the results are compared with theoretical predictions. In the first series of experiments, the downstream end of the clamped–clamped cylinder was free to slide axially, while in the second, the downstream end was fixed; the influence of externally applied axial compression was also studied in this series of experiments. The third series of experiments was similar to the second, except that a considerably more slender, hollow cylinder was used. In these experiments the cylinder lost stability by divergence at a sufficiently high flow velocity, and the amplitude of buckling increased thereafter. At higher flow velocities, the cylinder lost stability by flutter (attainable only in the third series of experiments), confirming experimentally the existence of a post-divergence oscillatory instability, which was previously predicted by both linear and nonlinear theory. Good quantitative agreement is obtained between theory and experiment for the amplitude of buckling, and for the critical flow velocities.
Dynamical systems and the transition to turbulence in linearly stable shear flows is studied by Eckhardt et al. (2008). It is known that plane Couette flow and pressure-driven pipe flow are two examples of flows where turbulence sets in while the laminar profile is still linearly stable. Experiments and numerical studies have shown that the transition has features compatible with the formation of a strange saddle rather than an attractor. In particular, the transition depends sensitively on initial conditions and the turbulent state is not persistent but has an exponential distribution of lifetimes. Embedded within the turbulent dynamics are coherent structures which transiently show up in the temporal evolution of the turbulent flow. The evidence for this transition scenario in these two flows, with an emphasis on lifetime studies in the case of plane Couette flow, and on the coherent structures in pipe flow, has been given.
In the paper by Hewitt et al. (2008) a nonlinear vortex development in rotating flows is shown where the steady secondary vortex flows is confined between two concentric right circular cylinders. When the flow is driven by the symmetric rotation of both end walls and the inner cylinder, toroidal vortex structures arise through the creation of stagnation points (in the meridional plane) at the inner bounding cylinder, or on the midplane of symmetry. A detailed description of the flow regimes is presented, suggesting that a cascade of such vortices can be created. Experimental results are reported that visualize some of the new states and confirm the prediction that they are stable to (midplane) symmetry breaking perturbations. Some brief results for flows driven by the rotation of a single end wall are also presented. Vortex structures may also be observed at low Reynolds numbers in this geometry. It is shown that standard flow visualization methods lead to some interesting non-axisymmetric particle paths in this case.
The final paper in this issue and the theme is written by Kobine (2008) on nonlinear resonant characteristics of shallow fluid layers. The presented results are from an experimental study of the motion of a shallow layer of water in a square tank that was oscillated horizontally with small amplitude at frequencies close to the natural frequency of the layer. The aim was to assess the validity of certain theoretical predictions relating to nonlinear resonance in shallow layers based on asymptotic analysis. These concern the hysteretic nature of the resonance profile and the existence of non-trivial oscillatory components associated with shock-like discontinuities in the derived solution for the free surface configuration. The experimental results support the theoretical predictions and show a coincidence in parameter space between the regions of hysteresis and oscillatory transition.
In closing this preface, I would like to express my thanks to the Editor, Prof. J. M. T. Thompson FRS for his support and encouragement. Of course, I am deeply grateful to all the authors and Dr Helen Ross, the publishing editor, for her patience and flexibility. At the end, I would like to reiterate the growing importance of nonlinear problems as more and more nonlinearities or nonlinear interactions are being deliberately used in science and technology.
One contribution of 6 to a Theme Issue ‘Experimental nonlinear dynamics II. Fluids’.
- © 2007 The Royal Society