Absolute instability in the near field of low-density jets

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

Non-Boussinesq stability analysis of natural-convection gaseous flow on inclined hot plates

The buoyancy-driven boundary-layer flow that develops over a semi-infinite inclined hot plate is known to become unstable at a finite distance from the leading edge, characterized by a critical value of the Grashof number Gr based on the local boundary-layer thickness. The nature of the resulting instability depends on the inclination angle /, measured from the vertical direction. For values of / below a critical value /c the instability is characterized by the appearance of spanwise traveling waves, whereas for/ > /c the bifurcated flow displays Görtler-like streamwise vortices. The Boussinesq approximation, employed in previous linear stability analyses, ceases to be valid for gaseous flow when the wall-to- ambient temperature ratio Hw is not close to unity. The corresponding non-Boussinesq analysis is pre- sented here, accounting also for the variation with temperature of the different transport properties. A temporal stability analysis including nonparallel effects of the base flow is used to determine curves of neutral stability, which are then employed to delineate the dependences of the critical Grashof number and of its associated wave length on the inclination angle / and on the temperature ratio Hw for the two instability modes, giving quantitative information of interest for configurations with Hw 1 1. The analysis provides in particular the predicted dependence of the crossover inclination angle /c on Hw , indicating that for gaseous flow with Hw 1 1 spanwise traveling waves are predominant over a range of inclination angles 0 6 / 6 /c that is significantly wider than that predicted in the Boussinesq approximation

Lubrication analysis of peristaltic motion in non-axisymmetric annular tubes

This paper addresses peristaltic flow induced in a non-axisymmetric annular tube by a periodic small-amplitude wave of arbitrary shape propagating axially along its inner surface, assumed to be a circular cylinder. The study is motivated by recent in vivo experimental observations pertaining to the flow of cerebrospinal fluid along the perivascular spaces of cerebral arteries. The analysis employs the lubrication approximation, describing low-Reynolds-number peristaltic flow in the long-wavelength approximation. Closed-form analytic expressions are derived for the average pumping rate in infinitely long tubes and also in tubes of finite length. Consideration is also given to the transverse motion arising in non-axisymmetric tubes. For small-amplitude waves, the solution is reduced to the integration of a parameter-free Stokes-flow problem, which is solved for relevant cross-sectional shapes, with closed-form analytical results derived for thin canals

Viscoacoustic squeeze-film force on a rigid disk undergoing small axial oscillations

This paper investigates the air flow induced by a rigid circular disk or piston vibrating harmonically along its axis of symmetry in the immediate vicinity of a parallel surface. Previous attempts to characterize these so-called 'squeeze-film' systems largely relied on simplifications afforded by neglecting either fluid acceleration or viscous forces inside the thin enclosed gas layer. The present viscoacoustic analysis employs the asymptotic limit of small vibration amplitudes to investigate the flow by systematic reduction of the Navier-Stokes equations in two distinct flow regions, namely, the inner gaseous film where streamlines are nearly parallel to the confining walls and the near-edge region of non-slender flow that features gas exchange with the surrounding stagnant atmosphere. The flow in the gaseous film depends on the relevant Stokes number, defined as the ratio of the characteristic viscous time across the film to the characteristic oscillation time, and on a compressibility parameter, defined as the square of the ratio of the acoustic time for radial pressure equilibration to the oscillation time. A Strouhal number based on the local residence time emerges as an additional governing parameter for the near-edge region, which is incompressible at leading order. The method of matched asymptotic expansions is used to describe the solution in both regions, across which the time-averaged pressure exhibits comparable variations that give opposing contributions to the resulting time-averaged force experienced by the disk or piston. A diagram structured with the Stokes number and compressibility parameter as coordinates reveals that this steady squeeze-film force, typically repulsive for small values of the Stokes number, alternates to attraction across a critical separation contour in the parametric domain that exists for all Strouhal numbers. This analysis provides, for the first time, a unifying viscoacoustic theory of axisymmetric squeeze films, which yields a reduced parametric description for the time-averaged repulsion/attraction force that is potentially useful in applications including non-contact fluid bearings and robot locomotion

Aerodynamics of planar counterflowing jets

The planar laminar flow resulting from the impingement of two gaseous jets of different density issuing into an open space from aligned steadily fed slot nozzles of semi-width H separated by a distance 2L is investigated by numerical and analytical methods. Specific consideration is given to the high Reynolds and low Mach number conditions typically present in counterflow-flame experiments, for which the flow is nearly inviscid and incompressible. It is shown that introduction of a density-weighted vorticity–streamfunction formulation effectively reduces the problem to one involving two jets of equal density, thereby removing the vortex-sheet character of the interface separating the two jet streams. Besides the geometric parameter L/H, the solution depends only on the shape of the velocity profiles in the feed streams and on the jet momentum-flux ratio. While conformal mapping can be used to determine the potential solution corresponding to uniform velocity profiles, numerical integration is required in general to compute rotational flows, including those arising with Poiseuille velocity profiles, with simplified solutions found in the limits L/H 1 and L/H 1. The results are used to quantify the near-stagnation-point region, of interest in counterflow-flame studies, including the local value of the strain rate as well as the curvature of the separating interface and the variations of the strain rate away from the stagnation point.This research was funded by the US AFOSR grant no. FA9550-16-1-0321. The inputs of Professor S. L. Smith, J. Carpio, J. C. Lasheras, A. Liñán, and F. A. Williams on different aspects of this research are gratefully acknowledged

Viscous stability analysis of jets with discontinuous base profiles

The viscous linear stability of parallel gaseous jets with piecewise constant base profiles is considered in the limit of low Mach numbers. Our results generalise those of Drazin [P.G. Drazin, Discontinuous velocity profiles for the Orr–Sommerfeld equation J. Fluid Mech. 10 (1961) 571–583], 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 linearised conservation equations in their primitive variable form. The development leads to an algebraic dispersion relation of ample validity that explicitly accounts for the parametric dependence of the stability properties on the jet-to-ambient density ratio, the Reynolds number, the Prandtl number, and the exponent of the presumed power-law dependence of viscosity and thermal conductivity on temperature. The dispersion relation 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 non-isothermal gaseous jets with very thin shear layers. The eigenvalue computations performed by using the vortex-sheet model are shown to be several orders of magnitude faster than those associated with continuous profiles with thin shear layers.This work was supported by Spanish MCINN through the project CONSOLIDER #CSD2010-00010, and the projects #DPI2011-28356-C03-02 and #ENE2008-06515-C04-01, and by the Comunidad de Madrid through projects #S2009/ENE-1597 and #CCG10-UC3M/DPI-4777

A model for the oscillatory flow in the cerebral aqueduct

This paper addresses the pulsating motion of cerebrospinal fluid in the aqueduct of Sylvius, a slender canal connecting the third and fourth ventricles of the brain. Specific attention is given to the relation between the instantaneous values of the flow rate and the interventricular pressure difference, needed in clinical applications to enable indirect evaluations of the latter from direct magnetic resonance measurements of the former. An order of magnitude analysis accounting for the slenderness of the canal is used in simplifying the flow description. The boundary layer approximation is found to be applicable in the slender canal, where the oscillating flow is characterized by stroke lengths comparable to the canal length and periods comparable to the transverse diffusion time. By way of contrast, the flow in the non-slender opening regions connecting the aqueduct with the two ventricles is found to be inviscid and quasi-steady in the first approximation. The resulting simplified description is validated by comparison with results of direct numerical simulations. The model is used to investigate the relation between the interventricular pressure and the stroke length, in parametric ranges of interest in clinical applications.The work of A.L.S. was supported by the National Science Foundation through grant no. 1853954. The work of W.C. was supported by the Convenio Plurianual Comunidad de Madrid Universidad Carlos III de Madrid' through grant no. CSFLOW-CM-UC3M

Observed dependence of characteristics of liquid-pool fires on swirl magnitude

One dozen vertically oriented thin rectangular vanes, 62 cm tall and 15.2 cm wide, were placed 27 cm from the center of heptane and ethanol pool fires in continuously fed, floor-flush pans 3.2 cm and 5.1 cm in diameter in the laboratory. The vanes were all oriented at the same fixed angles from the radial direction, for 9 different angles, ranging from 0 degrees to 85 degrees, thereby imparting 9 different levels of circulation to the air entrained by each pool fire. The different swirl levels were observed to engender dramatically different pool-fire structures. Moderate swirl suppresses the global puffing instability, replacing it by a global helical instability that generates a tall fire whirl, the height of which increases with increasing circulation. Except for the largest heptane pool, higher swirl levels produced vortex breakdown, resulting in the emergence of a bubble-like recirculation region with a ring vortex encircling the axis. Measured burning rates increase with increasing swirl levels as a consequence of the associated increasing inflow velocities reducing the thickness of the boundary layer within which combustion occurs right above the liquid surface, eventually forming detached edge flames in the boundary layer that move closer to the axis as the circulation is increased. Still higher circulation reduces the burning rate by decreasing the surface area of the liquid covered by the flame, thereby reducing the height of the fire whirl. Even higher circulation causes edge-flame detachment, resulting in formation of the blue whirl identified in recent literature, often meandering over the surface of the liquid in the present experiments. This sequence of events is documented herein

Oscillating viscous flow past a streamwise linear array of circular cylinders

This paper addresses the viscous flow developing about an array of equally spaced identical circular cylinders aligned with an incompressible fluid stream whose velocity oscillates periodically in time. The focus of the analysis is on harmonically oscillating flows with stroke lengths that are comparable to or smaller than the cylinder radius, such that the flow remains two-dimensional, time-periodic and symmetric with respect to the centreline. Specific consideration is given to the limit of asymptotically small stroke lengths, in which the flow is harmonic at leading order, with the first-order corrections exhibiting a steady-streaming component, which is computed here along with the accompanying Stokes drift. As in the familiar case of oscillating flow over a single cylinder, for small stroke lengths, the associated time-averaged Lagrangian velocity field, given by the sum of the steady-streaming and Stokes-drift components, displays recirculating vortices, which are quantified for different values of the two relevant controlling parameters, namely, the Womersley number and the ratio of the inter-cylinder distance to the cylinder radius. Comparisons with results of direct numerical simulations indicate that the description of the Lagrangian mean flow for infinitesimally small values of the stroke length remains reasonably accurate even when the stroke length is comparable to the cylinder radius. The numerical integrations are also used to quantify the streamwise flow rate induced by the presence of the cylinder array in cases where the periodic surrounding motion is driven by an anharmonic pressure gradient, a problem of interest in connection with the oscillating flow of cerebrospinal fluid around the nerve roots located along the spinal canal.This work was supported by the National Institute of Neurological Disorders and Stroke through contract no. 1R01NS120343-01 and by the National Science Foundation through grant no. 1853954. The work of W.C. was partially supported by the Spanish MICINN through the coordinated project PID2020-115961RB

Floquet stability analysis of a two-layer oscillatory flow near a flexible wall

We investigate the linear Floquet stability of two fluid layers undergoing oscillations in the direction parallel to the flexible wall that separates them. This canonical configuration is inspired by the cerebrospinal fluid flow in the spinal canal of subjects with hydro-/syringomyelia.The analysis focuses on the marginal conditions for the onset of instability, and how these depend on the spatial wavelength of the perturbation, and on the values of the control parameters, which are the two channel widths, the Reynolds number, and the wall stiffness. Unstable perturbations are found to oscillate synchronous with the base flow. The wavelength of the most unstable perturbation, of the order of the stroke length of the basic oscillatory motion, depends strongly on the wall stiffness, but is only weakly influenced by the channel widths and the Reynolds number. In general, around criticality, it was found that increasing the Reynolds number has a destabilizing effect, and that decreasing the canal widths stabilizes the instability. The wall stiffness on the other hand has a non-monotonic effect, exhibiting an intermediate value for which the instability is maximally amplified. The present analysis is a first step towards a better understanding of the physical mechanisms that govern many (bio)fluid mechanical problems that involve oscillatory flows near compliant walls