# Order in chaos

The following series of citations gives a sense of the impression that this discovery made on researchers working in the field of both turbulence and aeroacoustics.

“The apparently intimate connexion between jet stability and noise generation appears worthy of further investigation” – Mollo-Christensen and Narasimha (1960)

“[jet noise] is of interest as a problem in fluid dynamics in the class of problems which involve the interaction between instability, turbulence and wave emission” – Mollo-Christensen (1963)

“There appear to be at least two distinguishable types of emitted sound, one dominating at very low frequencies and another dominating at high frequencies. A relation which gives a smooth interpolation between these asymptotic ranges would prove useful, if one could be invented.” – Mollo – Christensen (1963)

“The data suggest that one may perhaps represent the fluctuating [hydrodynamic] pressure field in terms of rather simple functions. For example, one may consider the jet as a…semi-infinite antenna for sound…” – Mollo – Christensen (1967)

“…although the velocity signal is random, one should expect to see intermittently a rather regular spatial structure in the shear layer.” – Mollo – Christensen (1967)

“We therefore decided to stress measurements near and in the jets, hoping to discern some of the simpler features of the turbulent field. We also did measure for field pressures, and intended to see if we could not connect the two sets of observations somehow, using the equations of sound propagation.” – Mollo-Christensen (1967)

“It is suggested that turbulence, at least as far as some of the lower order statistical measures are concerned, may be more regular than we may think it is, if we could only find a new way of looking at it.” – Mollo-Christensen (1967)

“The mechanics of turbulence remains obscure, so that it comes as a matter of some relief to find that the motions which now interest us are coherent on a large scale…Such large eddies might be readily recognisable as a coherent transverse motion more in the category of a complicated laminar flow than chaotic turbulence. In any event the eddies generating the noise seem to be much bigger than those eddies which have been the subject of intense turbulence study. They are very likely those large eddies which derive their energy from an instability of the mean motion…” – Bishop et al. (1971)

“These [measurements] suggest that hidden in the apparently random fluctuations in the mixing layer region is perhaps a very regular and ordered pattern of flow which has not been detected yet” – Fuchs (1972)

“Whether one views these structures as waves or vortices is, to some extent, a matter of viewpoint.” – Brown and Roshko (1974)

“All this evidence suggests that the turbulence in the mixing layer of the jet behaves like a train similar to the hydrodynamic stability waves propagating in the shear flow.” – Chan (1974)

“The dominant role of the dynamics and interaction of the large structure in the overall mechanism that eventually brings the two fluids into intimate contact becomes apparent. It is clear that any theoretical attempts to model the complex mixing process in the shear layer must take this ubiquitous large structure into account.” – Dimotakis and Brown (1976)

“Turbulence research has advanced rapidly in the last decade with the widespread recognition of orderly large-scale structure in many kinds of turbulent shear flows…some measure of agreement seems to have been reached among investigators on the general properties of the coherent motions.” – Crighton and Gaster (1976)

“…the turbulence establishes an equivalent laminar flow profile as far as large-scale modes are concerned.” – Crighton and Gaster (1976)

“In the last years our understanding of turbulence, especially in jets, has changed rather dramatically. The reason is that jet turbulence has been found to be more regular than had been thought before.” – Michalke (1977)

“This ‘new-look’ in shear-flow turbulence, contrary to the classical notion of essentially complete chaos and randomness, has engendered an unusually high contemporary interest in the large-scale structures.” – Hussain and Zaman (1981)

The last twenty years of research on turbulence have seen a growing

realisation that the transport properties of most turbulence shear flows are dominated by large-scale vortex motions that are not random.” – Cantwell (1981)

“Suddenly it was feasible and reasonable to draw a picture of turbulence! The hand, the eye, and the mind were brought into a new relationship that had never quite existed before; cartooning became an integral part of the study of turbulence.” – Cantwell (1981)

As we see from many of the above citations, stability theory is frequently evoked as a possible theoretical framework for the dynamical modelling of the flow behaviour described above. However, a full treatment of hydrodynamic stability is beyond the scope of this lecture, and so we will simply list, briefly, a few of the different kinds of stability frameworks that are sometimes used to model the organised component of turbulent shear flows. We would also point out that the application of stability theory to turbulent flows, where the stability of a time-averaged mean-flow is considered, is not entirely rigorous (hydrodynamic stability analysis is self-consistent only when applied to laminar flows), involving a number of assumptions: one of these is that there exists a scale-separation between a large-scale organised component of the flow and a finer-grained, stochastic, ‘background’ component; the latter establishes a mean-flow profile that can sustain large-scale instabilities, and acts, furthermore, as a kind of eddy viscosity that damps the large-scale instabilities.

The first stability calculations with respect to the round jet were performed by Batchelor and Gill (1962) who studied the temporal stability problem for a plug flow. Michalke and Timme (1967) looked at the temporal instability of a finite-thickness, two-dimensional shear layer, while Michalke (1971) considered the spatial instability of a finite thickness axisymmetric shear-layer. Crighton & Gaster (1976) took account of the slow axial variation of the shear-layer thickness. Mankbadi and Liu (1984) made an attempt to include the effect of non-linearities. Tam and Morris (1980) used matched asymptotic expansions to obtain the acoustic field of a two-dimensional compressible mixing-layer; Tam and Burton (1984) then extending this effort to the case of a round jet. More recent approaches have been based on linear and non-linear Parabolised Stability Equations, as used by Colonius et al. (2010) for example, and Global Stability approaches, applied for instance to the problem of heated jets by Lesshafft et al. (2010).

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