Publication: Absolute instability in the near field of low-density jets
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2010-04
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2010-05-21
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Abstract
Variable density jets are known to support self-sustained oscillations
when the jet-to-ambient density ratio is sufficiently small. This
change in dynamical response to small perturbations is associated
with a transition from convective to absolute instability of the underlying
unperturbed base flow. The focus of this dissertation lies in the
use of linear stability theory to describe the convective to absolute
instability transition of buoyancy-free low-density jets emerging
from a circular injector tube at moderately high Reynolds numbers
and low Mach numbers. Particular interest is given to the in- fluence
of the length of the injector tube on the stability characteristics
of the resulting jet flow, whose base velocity profile at the jet
exit is computed in terms of the nondimen- sional tube length L$_{t}$
by integrating the boundary layer equations along the injector. We
begin with the investigation of inviscid axisymmetric and helical
modes of in- stability in a heated jet for different values of the
jet-to-ambient density ratio. For short tubes L$_{t}$ $\ll$ 1 the
base velocity profile at the tube exit is uniform except in a thin
sur- rounding boundary layer. Correspondingly, the stability analysis
reproduces previous results of uniform velocity jets, according to
which the jet becomes absolutely unstable to axisymmetric modes for
a critical density ratio S$_{c}$ $\simeq$ 0.66, and to helical modes
for S$_{c}$ $\simeq$ 0.35. For tubes of increasing length the analysis
reveals that both modes exhibit absolutely unstable regions for all
values of L$_{t}$ and small enough values of the density ratio. In
the case of the helical mode, we find that S$_{c}$ increases monotonically
with L$_{t}$ , reaching its maximum value S $\simeq$ 0.5 as the
exit velocity approaches the Poiseuille pro- file for L$_{t}$ $\gg$
1. Concerning the axisymmetric mode, its associated value of S$_{c}$
achieves a maximum value S$_{c}$ $\simeq$ 0.9 for $_{t}$ $\simeq$
0.04 and then decreases to approach S$_{c}$ $\simeq$ 0.7 for L$_{t}$
$\gg$ 1. The absolute growth rates in this limiting case of near-Poiseuille
jet profiles are however extremely small for m = 0, in agreement with
the fact that axisymmetric dis- turbances of a jet with parabolic
profile are neutrally stable. As a result, for S < 0.5 the absolute
growth rate of the helical mode becomes larger than that of the axisymmetric
mode for sufficiently large values of L$_{t}$ , suggesting that the
helical mode may prevail in the instability development of very light
jets issuing from long injectors. A second part of this dissertation
is devoted to the viscous linear instability of parallel gas flows
with piecewise constant base profiles in the limit of low Mach numbers,
both for planar and axisymmetric geometries such as mixing layers,
jets and wakes. Our results generalize those of Drazin (J. Fluid Mech.
vol. 10, 1961, p. 571), by contemplating the possibility of arbitrary
jumps in density and transport properties between two uniform streams
separated by a vortex sheet. The eigenfunctions, obtained analytically
in the regions of uniform flow, are matched through an appropriate
set of jump conditions at the discontinuity of the basic flow, which
are derived by repeated integration of the linearized conservation
equations in their primitive variable form. The development leads
to an algebraic dispersion relation that is validated through comparisons
with stability calculations performed with continuous profiles and
is applied, in particular, to study the effects of molecular transport
on the spatiotemporal stability of parallel nonisothermal gaseous
jets and wakes with very thin shear layers. Finally we go back to
the stability analysis of low-density jets emerging from circular
nozzles or tubes, this time considering viscous perturbations so that
the Reynolds number enters the stability problem. We consider separately
the two particular cases of a hot gas jet discharging into a colder
ambient of the same gas, as well as the isothermal discharge of a
jet of gas with molecular weight smaller than that of the ambient
gas. In both cases, we consider the detailed downstream evolution
of the local stability properties in the near field of the jet with
the aim at establishing the convective or absolute nature of the instability.
We discuss the relationship of our results with those obtained in
previous works with use made of parametric velocity and density profiles,
and compare both approaches with the actual global transition observed
in experiments performed with hot and light jets.
Description
Keywords
Low density jets, Linear stability theory, Chorros laminares, TeorÃa de estabilidad lineal