NONLINEAR DYNAMICS OF FORCED TRANSITIONAL JETS - PERIODIC AND CHAOTICATTRACTORS

Conclusive experimental evidence is presented for the existence of a l ow-dimensional temporal dynamical system in an open flow, namely the n ear field of an axisymmetric, subsonic free jet. An initially laminar jet (4 cm air jet in the Reynolds number range 1.1 x 10(4) less-than-o r-equal-to Re(D) less-than-or-equal-to 9.1 X 10(4) with a top-hat prof ile was studied using single-frequency, longitudinal, bulk excitation. Two non-dimensional control parameters - forcing frequency St(D) (=f( ex)D/U(e), where f(ex) is the excitation frequency, D is the jet exit diameter and U(e) is the exit velocity) and forcing amplitude a(f) (= u(f)'/U(e), where u(f)' is the jet exit r.m.s. longitudinal velocity f luctuation at the excitation frequency) - were varied over the ranges 10(-4) < a(f) < 0.3 and 0.3 < St(D) < 3.0 in order to construct a phas e diagram. Periodic and chaotic states were found over large domains o f the parameter space. The periodic attractors correspond to stable pa iring (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 exh ibits approximately quasi-periodic behaviour, including modulation sid ebands around 1/2f(ex) in u-spectra, large closed loops in its Poincar e sections, correlation dimension nu almost-equal-to 2 and largest Lya punov exponent lambda1 almost-equal-to 0. But investigations at smalle r scales (i.e. distances greater than, but of the order of, trajectory separation) in phase space reveal chaos, as shown by nu > 2 and lambd a1 > 0. The other chaotic attractor, near SDP, results from nearly per iodic modulations of the first vortex pairing but chaotic modulations of the second pairing and has a broadband spectrum, a dimension 2.5 le ss-than-or-equal-to nu less-than-or-equal-to 3 and the largest Lyapuno v exponent 0.2 less-than-or-equal-to lambda1 less-than-or-equal-to 0.7 bits per orbit (depending on measurement locations in physical and pa rameter spaces). A definition that distinguishes between physically an d dynamically open flows is proposed and justified by our experimental results. The most important conclusion of this study is that a physic ally open flow, even one that is apparently dynamically open due to co nvective instability, can exhibit dynamically closed behaviour as a re sult of feedback. A conceptual model for transitional jets is proposed based on two-dimensional instabilities, subharmonic resonance and fee dback from downstream vortical structures to the nozzle lip. Feedback was quantified and shown to affect the exit fundamental-subharmonic ph ase difference phi - a crucial variable in subharmonic resonance and, hence, vortex pairing. The effect of feedback, the sensitivity of pair ings to phi, the phase diagram, and the documented periodic and chaoti c attractors demonstrate the validity of the proposed conceptual model .