Perturbed Free Shear Layers

Jets, ie collimated streams of matter, occur from the microscale up to the large-scale structure of the universe Our focus will be mostly on surface tension effects, which result from the cohesive properties of liquids Paradoxically, cohesive forces promote the breakup of jets, widely encountered in nature, technology and basic science, for example in nuclear fission, DNA sampling, medical diagnostics, sprays, agricultural irrigation and jet engine technology Liquid jets thus serve as a paradigm for free-surface motion, hydrodynamic instability and singularity formation leading to drop breakup In addition to their practical usefulness, jets are an ideal probe for liquid properties, such as surface tension, viscosity or non-Newtonian rheology They also arise from the last but one topology change of liquid masses bursting into sprays Jet dynamics are sensitive to the turbulent or thermal excitation of the fluid, as well as to the surrounding gas or fluid medium The aim of this review is to provide a unified description of the fundamental and the technological aspects of these subjects

Physics of liquid jets

Experimental investigations of shear layer instability have shown that some obviously essential features of the instability properties cannot be described by the inviscid linearized stability theory of temporally growing disturbances. Therefore an attempt is made to obtain better agreement with experimental results by means of the inviscid linearized stability theory of spatially growing disturbances. Thus using the hyperbolic-tangent velocity profile the eigenvalues and eigenfunctions were computed numerically for complex wave-numbers and real frequencies. The results so obtained showed the tendency expected from the experiments. The physical properties of the disturbed flow are discussed by means of the computed vorticity distribution and the computed streaklines. It is found that the disturbed shear layer rolls up in a complicated manner. Furthermore, the validity of the linearized theory is estimated. The result is that the error due to the linearization of the disturbance equation should be larger for the vorticity distribution than for the velocity distribution, and larger for higher disturbance frequencies than for lower ones. Finally, it can be concluded from the comparison between the results of experiments and of both the spatial and temporal theory by Freymuth that the theory of spatially-growing disturbances describes the instability properties of a disturbed shear layer more precisely, at least for small frequencies.

On spatially growing disturbances in an inviscid shear layer

We depict and analyse the successive steps of atomization of a liquid jet when a fast gas stream blows parallel to its surface. Experiments performed with various liquids in a fast air flow show that the liquid destabilization proceeds from a two-stage mechanism: a shear instability first forms waves on the liquid. The transient acceleration experienced by the liquid suggests that a Rayleigh–Taylor type of instability is triggered at the wave crests, producing liquid ligaments which further stretch in the air stream and break into droplets.

On spray formation

Some of the consequences of the conservation of magnetic helicity for incompressible three-dimensional turbulent MHD flows are investigated. Absolute equilibrium spectra for inviscid infinitely conducting flows truncated at lower and upper wavenumbers kmin and kmax are obtained. When the total magnetic helicity approaches an upper limit given by the total energy (kinetic plus magnetic) divided by kmin, the spectra of magnetic energy and helicity are strongly peaked near kmin; in addition, when the cross-correlations between the velocity and magnetic fields are small, the magnetic energy density near kmin greatly exceeds the kinetic energy density. Several arguments are presented in favour of the existence of inverse cascades of magnetic helicity towards small wavenumbers leading to the generation of large-scale magnetic energy.

Possibility of an inverse cascade of magnetic helicity in magnetohydrodynamic turbulence

The stability of a free laminar layer between parallel streams is examined. The neutral curve and the curves of constant amplification are obtained by a numerical method for Reynolds numbers ranging from 0 to 40 as well as for the inviscid case. No minimum Reynolds number is found. The eigenfunctions are discussed. The case of a layer of increasing thickness is considered, and it is concluded that turbulent transition will occur when the Reynolds number reaches 150.

Stability of a Shear Layer between Parallel Streams

Semi‐Isotropic Turbulence and Helicoidal Flows

The statistics of the time evolution of a nonlinearly coupled system of first‐order equations representing the Euler equations is studied. The probability distribution of functions is nearly Gaussian, while that of their time derivatives has exponential tails and moments of order 4, 6, and 8 that approach those of the exponential distributions.

https://orbit.dtu.dk/files/4407390/larrsen.pdf

A non‐Gaussian model of turbulence (soccer‐ball integrals)

The inviscid flow produced by the initial conditions proposed by Taylor and Green illustrates some basic features of the turbulent flow process, namely, the build‐up of the mean square vorticity and the decay of the integral scale. The rate of deformation and the rate of vortex stretching along one direction shows some differences. The deformations occur mostly in lenticular regions. The evolution process of the flow development is discussed.

Numerical study of the Taylor–Green vortices

The principal assumptions used by Kraichnan in formulating a theory of turbulence are reviewed, and their meaning is examined in the light of a mathematical model. A regression function is defined and used to evaluate the triple correlations. The importance of the fourth‐order cumulants is discussed. The merits of the theory are illustrated by comparison with experimental results—in the case of the mathematical model—of grid turbulence (skewness factor) and of turbulence at very large Reynolds number.

https://apps.dtic.mil/sti/pdfs/AD0814983.pdf

Robert Betchov

Papers

Stability of a Shear Layer between Parallel Streams

Semi‐Isotropic Turbulence and Helicoidal Flows

A non‐Gaussian model of turbulence (soccer‐ball integrals)

Numerical study of the Taylor–Green vortices

Review of Kraichnan's Theory of Turbulence