DYNAMICS OF TURBULENT
FREE JETS
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, 
--
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/
),
where C is the local curvature of the tube,
is
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.
References
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).
