Fernando F. Grinstein
Laboratory for Computational Physics & Fluid Dynamics,Code 6410, Naval Research Laboratory, Washington, DC 20375-5344, USA (grinstein@LCP.NRL.NAVY.MIL)

Abstract The dynamics and topology of turbulent free jets are examined based on results of recent numerical simulations. Special attention is devoted to the case of jets emerging from non-circular nozzles, which are of great interest in practical applications because of their enhanced entrainment and mixing properties relative to those of comparable axisymmetric jets. The jet development is characterized by the dynamics of large scale vortex rings and braid vortices, and by strong vortex interactions leading to a more disorganized flow regime characterized by smaller-scale elongated vortex tubes. Results on the dynamics of isolated low aspect-ratio rectangular rings are used to obtain insights on vortex self-deformation and reconnection mechanisms involved in the transition to the turbulent jet regime.

Fluid dynamic jets and plumes constitute omnipresent phenomena in nature. Some of these phenomena are obvious to even the most casual observer -- like the jets which exit from one's mouth when exhaling on a cold morning, while others may require extraordinary efforts to be seen -- like astrophysical jets at distances of light years away visible only through telescopes. The jets of technology dominate our lives, from propelling the aircraft which move us across continents to providing the simple air hoses of the machine shop, from the stacks which spew the waste products of industry to the diffuser arrays which disperse effluent into our streams and rivers. Controlling the mixing of the jet (or plume) with its surroundings is the focus of active research. Typical practical applications demand enhanced combustion between injected fuel and background oxidizer, rapid initial mixing and submergence of effluent fluid, or improved mixing of a hot exhaust with the surroundings to affect afterburning. In this context, there is a crucial interest in recognizing and understanding the local nature of the jet instabilities and their global nonlinear development in space and time. On the other hand, purely academic research in this area is also exciting since it deals with the many challenging unresolved scientific issues of non-linear fluid dynamics and turbulence.
The jet dynamics is entirely controlled by the vorticity, VORTICITY EQUATION-- defined in terms of the velocity field u, which gives a local measure of the rotation of the fluid [1]. It is the generation of vorticity near the jet exit and its convection and diffusion into only a narrow portion of the total space available which gives a jet its defining appearance. The onset of flow instabilities and the resulting amplification of fluctuating vorticity, causes a further confinement of vorticity -- so much so that the characteristic narrow conical flow is easily visualized by a variety of means. The jet regime can be characterized by the jet exit velocity Uo, the jet diameter D, and the Reynolds number Re=UoD/v, where v is the kinematic viscosity. At moderately high Reynolds numbers, laminar jets are not stable, even in the absence of density variation and thermal effects, and rapidly develop into fully three-dimensional turbulent flows [2].
Mixing between a turbulent jet and its surroundings occurs in two stages, an initial stage of bringing relatively large amounts of the fluids together (large-scale stirring), and a second stage promoted by the small-scale velocity fluctuations which accelerate mixing at the molecular level. Especially important is the rate at which fluid from the jet and from its surroundings become entangled or mixed as they join at the mixing layers. This information is given by the entrainment rate of the jet, which defines the rate of propagation of the interface between rotational and irrotational fluid, controlled by the speed at which the interface contortions with the largest scales move into the surrounding fluid [2]. These controlling large scale vortices tend to be coherent and easily recognizable features, hence their name coherent structures (CS). Control of the jet development is strongly dependent on understanding the dynamics and topology of CS; in particular, how the jet properties can be affected through control of the formation, interaction, merging, and breakdown of CS.
In the simplest conceptual jet picture, that of the free laminar jet emerging from an axisymmetric nozzle, a shear layer is formed immediately downstream of the jet exit between jet stream and surroundings. As one moves downstream, there is an early linear instability jet regime, involving exponential growth of small perturbations introduced at the jet exit. Beyond this development stage, in the non-linear Kelvin-Helmholtz instability regime, large scale vortex rings roll up, and their dynamics of formation and merging become the defining feature of the transitional shear flow [3]. Laboratory studies show that azimuthal effects can be important fairly close to the jet exit [4], so that the simple axisymmetric vortex-ring picture of the circular jet becomes less realistic as one moves downstream from the jet exit. Sufficiently far downstream, three-dimensionality is the crucial feature of the jet and the streamwise vorticity component (along the jet axial direction) is the more efficient one in entraining fluid from the surroundings. In addition to the important role of azimuthal instabilities contributing to the breakdown of the vortex rings, mechanisms such as self-induction, vortex stretching, and vortex reconnection are the main fluid-dynamical processes involved in the transition to the turbulent jet regime [5]. The most important new element introduced in the jet by three-dimensional instabilities is that now in addition to the shear vorticity production, vorticity can now also be generated locally through other mechanisms (e.g., by stretching and turning of old vorticity), and there are interesting new possibilities for how the jet flow develops, depending on which types of unsteady vortex interactions are initiated.
Passive shear-flow control methods to enhance the three-dimensionality of the flow, and thus entrainment and mixing, were studied to manipulate the natural development of CS and their breakdown into turbulence. Passive mixing-control strategies are based on geometrical modifications of the jet nozzle which can directly alter the flow development downstream relative to using a conventional circular nozzle. Jet studies using non-axisymmetric nozzles show that as the jet spreads, its cross-section can evolve through shapes similar to that at the jet-exit but with axis succesively rotated at angles characteristic of the jet geometry (so-called axis-switching phenomena) [6]. Axis-Switching is the main mechanism behind the enhanced entrainment properties of non-circular jets relative to comparable circular jets. Numerical simulations elucidated the vortex dynamics underlying axis switching. Simulations of initially laminar square jets [7] showed that the basic jet development (Figure 1) is controlled by the dynamics of self-deforming vortex rings and hairpin (braid) vortices -- formed as a result of vortex induction and stretching processes in the high strain regions between rings. As we move downstream and the transition to turbulence takes place, strong interactions between rings and braid vortices, and azimuthal instabilities lead to more contorted vortices; vortex stretching, kinking, and reconnection lead to their breakdown, and to a more disorganized flow regime characterized by smaller-scale elongated (worm) vortices and spectral energy content consistent with the k-5/3 inertial subrange of Kolmogorov's cascade theory of turbulence (Figure 2).
The simulations show that the basic mechanism for the first (45o) axis-rotation of the jet cross-section is the self-induced Biot-Savart deformation of the vortex rings due to non-uniform azimuthal curvature at the initial jet shear layer (Figure 3). Subsequent axis-rotations of the jet cross-section result from the strong interactionbetween spanwise and streamwise vorticity.
Assuming incompressibility, inviscid flow conditions, and modelling a vortex ring in terms of a thin vortex tube, the azimuthal variation of the self-induced velocity u responsible for the vortex-ring deformations can be described by [8], u ~ C b log (1/SIGMA), where C is the local curvature of the tube, SIGMAis its local cross-section, and b is the binormal to the plane containing the tube. According to this expression, in the absence of azimuthal perturbations, a perfectly round uniform vortex ring will be convected without changes in shape. In contrast, for non-circular rings, vortex ring deformations will occur due to larger self-induced velocities at the portions of the ring where C is larger.
The self-deformation process for an isolated rectangular vortex ring [9] is shown in Figure 4for aspect ratio AR=2. In the first deformation phase, the corner regions move ahead faster due to the self-induced velocity; the higher-curvature portions left behind at major-axis locations then move faster and towards the jet centerline, and further self-induced deformation and vortex stretching lead to a vortex ring with formally switched axes. For AR<4, an isolated rectangular ring can undergo quite-regular self-induced axis-switchings approximatelyrecovering shape and flatness as shown in Figure 4.
Insight into some of the more complicated three-dimensional vortex dynamical processes in the jet can be obtained based on the study of vortex rings with AR>3, for which other interesting phenomena such as vortex reconnection, become possible. For AR=4, for example, the self-induced dynamics is first quite similar to that for AR=2 (Figure 5). However, as the ring moves downstream, the centers of its longer antiparallel portions approach each other, vortex reconnection takes place, and the ring eventually bifurcates into two smaller vortex rings (Figure 6) linked by very thin threads of vorticity [9]. Vortex-ring bifurcation has been observed in laboratory experiments with AR=4 jets, but the detailed vorticity redistribution leading to bridging of antiparallel portions of the vortex ring and formation of threads linking the split vortex rings cannot be captured by the experimental flow visualizations. The vortex bifurcation is followed by collision of the split rings and new reconnection processes leading to their fusion, accompanied by additional shedding of (predominantly axial) vorticity in its wake. The numerical simulations give direct evidence of cascade mechanisms for transition to turbulence based on succesive vortex reconnections.

1. F.F. Grinstein, M. Glauser, and W. K. George, Vorticity in Jets, Chapter III of Fluid Vortices, Ed. S. Green, pp. 65-94, Kluwer Academic Publishing (1995).

2. H. Tennekes and J.L. Lumley, A First Course in Turbulence, MIT Press (1972).

3. F.F.Grinstein, F. Hussain and E.S. Oran, Vortex-Ring Dynamics in a Transitional Subsonic Free Jet. A Numerical Study, Europ. J. Mech. B / Fluids, 9, 499 (1990).

4. A.J. Yule, Large-scale Structure in the Mixing Layer of a Round Jet, J. Fluid Mech., 89, 413 (1978).

5. A.K.M.F. Hussain, Coherent Structures and Turbulence, J. Fluid Mech., 173, 303 (1986).

6. F.F. Grinstein, E. Gutmark, and T. Parr, Near-Field Dynamics of Subsonic, Free Square Jets. A Computational and Experimental Study, Phys. Fluids, 7, 1483 (1995).

7. F.F. Grinstein and C.R. DeVore, Dynamics of Coherent Structures and Transition to Turbulence in Free Square Jets, Phys. Fluids, 8, 1237 (1996).

8. G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, p.510 (1967).

9. F.F. Grinstein, Self-Induced Vortex Ring Dynamics in Subsonic Rectangular Jets, Phys. Fluids, 7, 2519 (1995).