Past evidence suggests that a large-scale orderly pattern may exist in the noiseproducing region of a jet. Using several methods to visualize the flow of round subsonic jets, we watched the evolution of orderly flow with advancing Reynolds number. As the Reynolds number increases from order 102 to 103, the instability of the jet evolves from a sinusoid to a helix, and finally to a train of axisymmetric waves. At a Reynolds number around 104, the boundary layer of the jet is thin, and two kinds of axisymmetric structure can be discerned: surface ripples on the jet column, thoroughly studied by previous workers, and a more tenuous train of large-scale vortex puffs. The surface ripples scale on the boundary-layer thickness and shorten as the Reynolds number increases toward 105. The structure of the puffs, by contrast, remains much the same: they form at an average Strouhal number of about 0·3 based on frequency, exit speed, and diameter.To isolate the large-scale pattern at Reynolds numbers around 105, we destroyed the surface ripples by tripping the boundary layer inside the nozzle. We imposed a periodic surging of controllable frequency and amplitude at the jet exit, and studied the response downstream by hot-wire anemometry and schlieren photography. The forcing generates a fundamental wave, whose phase velocity accords with the linear theory of temporally growing instabilities. The fundamental grows in amplitude downstream until non-linearity generates a harmonic. The harmonic retards the growth of the fundamental, and the two attain saturation intensities roughly independent of forcing amplitude. The saturation amplitude depends on the Strouhal number of the imposed surging and reaches a maximum at a Strouhal number of 0·30. A root-mean-square sinusoidal surging only 2% of the mean exit speed brings the preferred mode to saturation four diameters downstream from the nozzle, at which point the entrained volume flow has increased 32% over the unforced case. When forced at a Strouhal number of 0·60, the jet seems to act as a compound amplifier, forming a violent 0·30 subharmonic and suffering a large increase of spreading angle. We conclude with the conjecture that the preferred mode having a Strouhal number of 0·30 is in some sense the most dispersive wave on a jet column, the wave least capable of generating a harmonic, and therefore the wave most capable of reaching a large amplitude before saturating.

/pdf/orderly-structure-in-jet-turbulence-1gxc8ijm65.pdf

Orderly Structure in Jet Turbulence

This paper is a contribution to the mathematical analysis of the stability of steady axisymmetric parallel flows of uniform fluid in the absence of rigid boundaries. A jet at sufficiently high Reynolds number for the angle of viscous spreding to be small is the typical example of the primary flows considered, and the theoretical velocity profile far downstream in such a jet is kept in mind continually. It is obvious from experience that such jets are unstable, presumably to infinitesimal disturbances, but there is little observational data about the critical Reynolds number or the mode of disturbance that grows most rapidly at a given Reynolds number.The typical small disturbance considered is a Fourier component with sinusoidal dependence on both ax and nϕ (x, r, ϕ are cylindrical polar co-ordinates). There is no analogue of Squire's theorem for two-dimensional primary flows, and both α and n are essential parameters of the disturbance. We have concentrated on the stability characteristics in the limit of large Reynolds number, and have aimed in particular at determining the (integral) value of n at which the growth rate is a maximum in these simpler circumstances.A number of general results for inviscid fluid are established, many of them analogues of corresponding results for two-dimensional primary flow. A necessary condition for the existence of amplified disturbances is that should have a numerical maximum a t some point in the fluid; this condition is satisfied for all n in the case of a cylindrical shear layer or ‘top-hat’ jet profile (for which a complete solution of the disturbance equation can be obtained), and for n > 1 in the case of a ‘far-downstream’ jet profile. The wave speed cr of a neutral disturbance is equal to the value of U either a t the point where dQ/dr = 0 or at r = 0. In the latter case the eigen-function (if one exists) is singular at the axis in general; the former case is presumably relevant to the ‘upper branch’ of the curve of neutral stability (for given n). The Reynolds stress due to the disturbance acting across a cylindrical surface is examined. Here, as in some other contexts, it is useful to consider components of velocity parallel and perpendicular to a circular helix on which the phase of the disturbance wave is constant. For a neutral disturbance the component of disturbance velocity parallel to the local wave helix is infinite a t the critical point where U = cr, (corresponding to the known singularity for a three-dimensional disturbance to two-dimensional flow), and there is a peak in the Reynolds stress there.It is shown from the form of the disturbance equation that there is an upper limit to the value of n (≠ 0) for a neutral (inviscid) disturbance with cr equal to the value of U at the point where Q′ = 0. In the case of a jet with a ‘far-downstream’ profile, only the value n = 1 satisfies this restriction; thus only the sinuous mode n = 1 can yield amplified disturbances in an inviscid fluid. A numerical investigation shows that for this profile the wave-number of the neutral disturbance with n = 1 is α = 1·46.

Analysis of the stability of axisymmetric jets

The stability of three axisymmetric jet profiles is reviewed. These profiles represent the development of an incompressible jet from a nearly top-hat profile to a fully developed jet profile. The disturbance equations for arbitrary mode number in a region of zero shear, which provide the boundary conditions for the numerical solution, are solved analytically through use of the disturbance vorticity equations. Numerical solutions for the spatial stability for the axisymmetric (n = 0) disturbance and the asymmetric n = l disturbance are presented. Previously published calculations of least stable modes are shown to be incorrectly interpreted and their actual mode types are given. The critical Reynolds number is found to increase as the profile varies from a top-hat to a fully developed jet form. Closed contours of constant amplification, which are unusual in free shear flows, are shown to exist for the n = 1 disturbance in the fully developed jet region. A fluctuation energy balance is used to justify the occurrence of this destabilizing effect of decreasing Reynolds number.

The spatial viscous instability of axisymmetric jets

The study of the fluid dynamics of violent volcanic eruptions by laboratory experiment is described, and the important fluid-dynamic processes that can be examined in laboratory models are discussed in detail. In preliminary experiments, pure gases are erupted from small reservoirs. The gases used are Freon 12 and Freon 22, two gases of high molecular weight and high density that are good analogs of heavy and particulate-laden volcanic gases; nitrogen, a moderate molecular weight, moderate density gas for which the thermodynamic properties are well known; and helium, a low molecular weight, lowdensity gas that is used as a basis for comparison with the behavior of the heavier gases and as an analog of steam, the gas that dominates many volcanic eruptions. Transient jets erupt from the reservoir into the laboratory upon rupture of a thin diaphragm at the exit of a convergent nozzle. The gas accelerates from rest in the reservoir to high velocity in the jet. Reservoir pressures and geometries are such that the fluid velocity in the jets is initially supersonic and later decays to subsonic. The measured reservoir pressure decreases as the fluid expands through repetitively reflecting rarefaction waves, but for the conditions of these experiments, a simple steady-discharge model is sufficient to explain the pressure decay and to predict the duration of the flow. Density variations in the flow field have been visualized with schlieren and shadowgraph photography. The observed structure of the jet is correlated with the measured pressure history. The starting vortex generated when the diaphragm ruptures becomes the head of the jet. Though the exit velocity is sonic, the flow head in the helium jet decelerates to about one-third of sonic velocity in the first few nozzle diameters, the nitrogen head decelerates to about three-fourths of sonic velocity, while Freon maintains nearly sonic velocity. The impulsive acceleration of reservoir fluid into the surrounding atmosphere produces a compression wave. The strength of this wave depends primarily on the sound speed of the fluid in the reservoir but also, secondarily with opposite effect, on the density: helium produces a relatively strong atmospheric shock while the Freons do not produce any optically observable wave front. Well-formed N waves are detected with a microphone far from the reservoir. Barrel shocks, Mach disks, and other familiar features of steady underexpanded supersonic jets form inside the jet almost immediately after passage of the flow head. These features are maintained until the pressure in the reservoir decays to sonic conditions. At low pressures the jets are relatively structureless. Gas-particle jets from volcanic eruptions may behave as pseudogases if particle concentrations and mass and momentum exchange between the components are sufficiently small. The sound speed of volcanic pseudogases can be as large as 1000 m s^(−1) or as small as a few tens of meters per second depending on the mass loading and initial temperature. Fluids of high sound speed produce stronger atmospheric shock waves than do those of low sound speed. Therefore eruption of a hot gas lightly laden with particulates should produce a stronger shock than eruption of a cooler or heavily laden fluid. An empirical expression suggests that the initial velocity of the head of supersonic volcanic jets is controlled by the sound speed and the ratio of the density of the erupting fluid to that of the atmosphere. The duration of gas or pseudogas eruptions is controlled by the sound speed of the fluid and the ratio of reservoir volume to vent area.

/pdf/laboratory-studies-of-volcanic-jets-4da48pmgi2.pdf

Laboratory studies of volcanic jets

Local heat transfer to impinging liquid jets in the initially laminar, transitional, and turbulent regimes

A jet of dyed water is directed into a large tank of water. As the flow rate is increased several successive modes of breakdown of the dyed column are observed:
 tiny puffs of dye near the nozzle;axisymmetric condensations well away from the nozzle;sinuous undulations of long wave length far from the nozzle;formation of foot-shaped pockets of dye;confused breakup near the nozzle with rapid spread of dye.
 The length of nearly rectilinear jet increases through the stages (a), (b) and (c). The mode (d) is an alternative to (c), occurring nearer the nozzle and leaving a shorter length of straight jet. Above a Reynolds number of about 300, only the confused breakup (e) is found.

A. J. Reynolds

Papers

Observations of a liquid-into-liquid jet