Conclusive experimental evidence is presented for the existence of a low-dimensional temporal dynamical system in an open flow, namely the near field of an axisymmetric, subsonic free jet. An initially laminar jet (4 cm air jet in the Reynolds number range 1.1 × 104 [Lt ] ReD × 9.1 × 104) with a top-hat profile was studied using single-frequency, longitudinal, bulk excitation. Two non-dimensional control parameters – forcing frequency StD (≡fexD/Ue, where fez is the excitation frequency, D is the jet exit diameter and Ue is the exit velocity) and forcing amplitude af (≡ u’f/Ue, where u’f is the jet exit r.m.s. longitudinal velocity fluctuation at the excitation frequency) – were varied over the ranges 10-4 < af < 0.3 and 0.3 < StD < 3.0 in order to construct a phase diagram. Periodic and chaotic states were found over large domains of the parameter space. The periodic attractors correspond to stable pairing (SP) and stable double pairing (SDP) of rolled-up vortices. One chaotic attractor, near SP in the parameter space, results from nearly periodic modulations of pairing (NPMP) of vortices. At large scales (i.e. approximately the size of the attractor) in phase space, NPMP exhibits approximately quasi-periodic behaviour, including modulation sidebands around ½fex in u-spectra, large closed loops in its Poincaré sections, correlation dimension v ∼ 2 and largest Lyapunov exponent λ1 ∼ 0. But investigations at smaller scales (i.e. distances greater than, but of the order of, trajectory separation) in phase space reveal chaos, as shown by v > 2 and λ1 > 0. The other chaotic attractor, near SDP, results from nearly periodic modulations of the first vortex pairing but chaotic modulations of the second pairing and has a broadband spectrum, a dimension 2.5 [Lt ] v [Lt ] 3 and the largest Lyapunov exponent 0.2 [Lt ] λ1 [Lt ] 0.7 bits per orbit (depending on measurement locations in physical and parameter spaces).A definition that distinguishes between physically and dynamically open flows is proposed and justified by our experimental results. The most important conclusion of this study is that a physically open flow, even one that is apparently dynamically open due to convective instability, can exhibit dynamically closed behaviour as a result of feedback. A conceptual model for transitional jets is proposed based on twodimensional instabilities, subharmonic resonance and feedback from downstream vortical structures to the nozzle lip. Feedback was quantified and shown to affect the exit fundamental–subharmonic phase difference ϕ – a crucial variable in subharmonic resonance and, hence, vortex pairing. The effect of feedback, the sensitivity of pairings to ϕ, the phase diagram, and the documented periodic and chaotic attractors demonstrate the validity of the proposed conceptual model.
Largest Lyapunov Exponents and Bifurcations of Stochastic Nonlinear Systems
