Absolute instability in the near field of low-density jets

Résumé

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.