Perturbed Free Shear Layers

A mixing layer is formed by bringing two streams of water, moving at different velocities, together in a lucite-walled channel. The Reynolds number, based on the velocity difference and the thickness of the shear layer, varies from about 45, where the shear layer originates, to about 850 at a distance of 50 cm. Dye is injected between the two streams just before they are brought together, marking the vorticity-carrying fluid. Unstable waves grow, and fluid is observed to roll up into discrete two-dimensional vortical structures. These turbulent vortices interact by rolling around each other, and a single vortical structure, with approximately twice the spacing of the former vortices, is formed. This pairing process is observed to occur repeatedly, controlling the growth of the mixing layer. A simple model of the mixing layer contains, as the important elements controlling growth, the degree of non-uniformity in the vortex train and the ‘lumpiness’ of the vorticity field.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A67DA3AA1031294B48E2216DB0CBF3D8/S0022112074001121a.pdf/div-class-title-vortex-pairing-the-mechanism-of-turbulent-mixing-layer-growth-at-moderate-reynolds-number-div.pdf

Vortex pairing : the mechanism of turbulent mixing-layer growth at moderate Reynolds number

This is a personal statement on the present state of understanding of coherent structures, in particular their spatial details and dynamical significance. The characteristic measures of coherent structures are discussed, emphasizing coherent vorticity as the crucial property. We present here a general scheme for educing structures in any transitional or fully turbulent flow. From smoothed vorticity maps in convenient flow planes, this scheme recognizes patterns of the same mode and parameter size, and then phase-aligns and ensemble-averages them to obtain coherent structure measures. The departure of individual realizations from the ensemble average denotes incoherent turbulence. This robust scheme has been used to educe structures from velocity data using a rake of hot wires as well as direct numerical simulations and can educe structures using newer measurement techniques such as digital image processing. Our recent studies of coherent structures in several free shear flows are briefly reviewed. Detailed data in circular and elliptic jets, mixing layers, and a plane wake reveal that incoherent turbulence is produced at the ‘saddles’ and then advected to the ‘centres’ of the structures. The mechanism of production of turbulence in shear layers is the stretching of longitudinal vortices or ‘ribs’ which connect the predominantly spanwise ‘rolls’; the ribs induce spanwise contortions of rolls and cause mixing and dissipation, mostly at points where they connect with rolls. We also briefly discuss the role of coherent structures in aerodynamic noise generation and argue that the structure breakdown process, rather than vortex pairing, is the dominant mechanism of noise generation. The ‘cut-and-connect’ interaction of coherent structures is proposed as a specific mechanism of aerodynamic noise generation, and a simple analytical model of it shows that it can provide acceptable predictions of jet noise. The coherent-structures approach to turbulence, apart from explaining flow physics, has also enabled turbulence management via control of structure evolution and interactions. We also discuss some new ideas under investigation: in particular, helicity as a characteristic property of coherent structures.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112086001192

Coherent structures and turbulence.

x(/3) Trailing vortices do not decay by simple diffusion. Usually they undergo a symmetric and nearly sinusoidal instability, until eventually they join at intervals to form a train of vortex rings. The present theory accounts for the instability during the early stages of its growth. The vortices are idealized as interacting lines; their core diameters are taken into account by a cutoff in the line integral representing self-induction. The equation relating induced velocity to vortex displacement gives rise to an eigenvalue problem for the growth rate of sinusoidal perturbations. Stability is found to depend on the products of vortex separation 6 and cutoff distance d times the perturbation wavenumber. Depending on those products, both symmetric and antisymmetric eigenmodes can be unstable, but only the symmetric mode involves strongly interacting long waves. An argument is presented that d/b = 0.063 for the vortices trailing from an elliptically loaded wing. In that case, the maximally unstable long wave has a length 8.66 and grows by a factor e in a time 9.4(^4#/CL)(6/F0), where AR is the aspect ratio, CL is the lift coefficient, and V0 is the speed of the aircraft. The vortex displacements are symmetric and are confined to fixed planes inclined at 48° to the horizontal.

Stability theory for a pair of trailing vortices

In the present study, it is shown that the spreading rate of a mixing layer can be greatly manipulated at very low forcing level if the mixing layer is perturbed near a subharmonic of the most-amplified frequency. The subharmonic forcing technique is able to make several vortices merge simultaneously and hence increases the spreading rate dramatically. A new mechanism, ‘collective interaction’, was found which can bypass the sequential stages of vortex merging and make a large number of vortices (ten or more) coalesce.A deeper physical insight into the evolution of the coherent structures is revealed through the investigation of a forced mixing layer. The stability and the forcing function play important roles in determining the initial formation of the vortices. The subharmonic starts to amplify at the location where the phase speed of the subharmonic matches that of the fundamental. The position where vortices are seen to align vertically coincides with the position where the measured subharmonic reaches its peak. This location is defined as the merging location, and it can be determined from the feedback equation (Ho & Nosseir 1981).The spreading rate and the velocity profiles of the forced mixing layer are distinctly different from the unforced case. The data show that the initial condition has a longlasting effect on the development of the mixing layer.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112082001438

Subharmonics and vortex merging in mixing layers

Past evidence suggests that a large-scale orderly pattern may exist in the noiseproducing region of a jet. Using several methods to visualize the flow of round subsonic jets, we watched the evolution of orderly flow with advancing Reynolds number. As the Reynolds number increases from order 102 to 103, the instability of the jet evolves from a sinusoid to a helix, and finally to a train of axisymmetric waves. At a Reynolds number around 104, the boundary layer of the jet is thin, and two kinds of axisymmetric structure can be discerned: surface ripples on the jet column, thoroughly studied by previous workers, and a more tenuous train of large-scale vortex puffs. The surface ripples scale on the boundary-layer thickness and shorten as the Reynolds number increases toward 105. The structure of the puffs, by contrast, remains much the same: they form at an average Strouhal number of about 0·3 based on frequency, exit speed, and diameter.To isolate the large-scale pattern at Reynolds numbers around 105, we destroyed the surface ripples by tripping the boundary layer inside the nozzle. We imposed a periodic surging of controllable frequency and amplitude at the jet exit, and studied the response downstream by hot-wire anemometry and schlieren photography. The forcing generates a fundamental wave, whose phase velocity accords with the linear theory of temporally growing instabilities. The fundamental grows in amplitude downstream until non-linearity generates a harmonic. The harmonic retards the growth of the fundamental, and the two attain saturation intensities roughly independent of forcing amplitude. The saturation amplitude depends on the Strouhal number of the imposed surging and reaches a maximum at a Strouhal number of 0·30. A root-mean-square sinusoidal surging only 2% of the mean exit speed brings the preferred mode to saturation four diameters downstream from the nozzle, at which point the entrained volume flow has increased 32% over the unforced case. When forced at a Strouhal number of 0·60, the jet seems to act as a compound amplifier, forming a violent 0·30 subharmonic and suffering a large increase of spreading angle. We conclude with the conjecture that the preferred mode having a Strouhal number of 0·30 is in some sense the most dispersive wave on a jet column, the wave least capable of generating a harmonic, and therefore the wave most capable of reaching a large amplitude before saturating.

/pdf/orderly-structure-in-jet-turbulence-1gxc8ijm65.pdf

S. C. Crow

Papers

Orderly Structure in Jet Turbulence