Structure Of Turbulent Jets Research Articles

Transition to chaos in a reduced-order model of a shear layer

The present work studies the nonlinear dynamics of a shear layer, driven by a body force and confined between parallel walls, a simplified setting to study transitional and turbulent shear layers. It was introduced by Nogueira & Cavalieri (J. Fluid Mech., vol. 907, 2021, A32), and is here studied using a reduced-order model based on a Galerkin projection of the Navier–Stokes system. By considering a confined shear layer with free-slip boundary conditions on the walls, periodic boundary conditions in streamwise and spanwise directions may be used, simplifying the system and enabling the use of methods of dynamical systems theory. A basis of eight modes is used in the Galerkin projection, representing the mean flow, Kelvin–Helmholtz vortices, rolls, streaks and oblique waves, structures observed in the cited work, and also present in shear layers and jets. A dynamical system is obtained, and its transition to chaos is studied. Increasing Reynolds number $Re$ leads to pitchfork and Hopf bifurcations, and the latter leads to a limit cycle with amplitude modulation of vortices, as in the direct numerical simulations by Nogueira & Cavalieri. Further increase of $Re$ leads to the appearance of a chaotic saddle, followed by the emergence of quasi-periodic and chaotic attractors. The chaotic attractors suffer a merging crisis for higher $Re$ , leading to a chaotic dynamics with amplitude modulation and phase jumps of vortices. This is reminiscent of observations of coherent structures in turbulent jets, suggesting that the model represents a dynamics consistent with features of shear layers and jets.

Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets

Abstract

Numerical simulations of high frequency transverse pulsed jet injection into a supersonic crossflow

The design of scramjet engines includes the investigation of minimally intrusive fuel delivery mechanisms which maximize jet fuel penetration and mixing. In subsonic crossflows it has been shown that pulsation of gaseous jets can improve jet penetration and mixing in comparison to unforced (steady) jets. Extensive investigations have already explored subsonic flow regimes for pulsed injection; however, only a limited number of experimental and numerical studies have been performed which investigate pulsed jets in a supersonic crossflow. This study presents fully three-dimensional (3D) scale-resolving simulations of steady and a sinusoidally pulsed jet (f = 16 kHz) using a wall modeled large eddy simulation approach to capture large-scale vortical structures in turbulent jet in supersonic crossflow and pulsed jet in supersonic crossflow. A block-structured, fully 3D turbulent scale resolving finite volume model was used to explore physics simulations of steady and pulsed jets in a supersonic crossflow. The results of the simulations presented show that sinusoidal pulsation of a gaseous hydrogen jet improves penetration and mixing over the steady jet when the momentum flux ratios are equivalent.

Large-scale streaky structures in turbulent jets

Streaks have been found to be an important part of wall-turbulence dynamics. In this paper, we extend the analysis for unbounded shear flows, in particular a Mach 0.4 round jet, using measurements taken using dual-plane, time-resolved, stereoscopic particle image velocimetry (PIV) taken at pairs of jet cross-sections, allowing the evaluation of the cross-spectral density of streamwise velocity fluctuations resolved into azimuthal Fourier modes. From the streamwise velocity results, two analyses are performed: the evaluation of wavenumber spectra (assuming Taylor’s hypothesis for the streamwise coordinate) and a spectral proper orthogonal decomposition (SPOD) of the velocity field using PIV planes in several axial stations. The methods complement each other, leading to the conclusion that large-scale streaky structures are also present in turbulent jets where they experience large growth in the streamwise direction, energetic structures extending up to eight diameters from the nozzle exit. Leading SPOD modes highlight the large-scale, streaky shape of the structures, whose aspect ratio (streamwise over azimuthal length) is approximately 15. The data were further analysed using SPOD, resolvent and transient growth analyses, good agreement being observed between the models and the leading SPOD mode for the wavenumbers considered. The models also indicate that the lift-up mechanism is active in turbulent jets, with streamwise vortices leading to streaks. The results show that large-scale streaks are a relevant part of the jet dynamics.

Wave-Packet Models for Jet Dynamics and Sound Radiation

Organized structures in turbulent jets can be modeled as wavepackets. These are characterized by spatial amplification and decay, both of which are related to stability mechanisms, and they are coherent over several jet diameters, thereby constituting a noncompact acoustic source that produces a distinctive directivity in the acoustic field. In this review, we use simplified model problems to discuss the salient features of turbulent-jet wavepackets and their modeling frameworks. Two classes of model are considered. The first, that we refer to as kinematic, is based on Lighthill's acoustic analogy, and allows an evaluation of the radiation properties of sound-source functions postulated following observation of jets. The second, referred to as dynamic, is based on the linearized, inhomogeneous Ginzburg–Landau equation, which we use as a surrogate for the linearized, inhomogeneous Navier–Stokes system. Both models are elaborated in the framework of resolvent analysis, which allows the dynamics to be viewed in terms of an input–ouput system, the input being either sound-source or nonlinear forcing term, and the output, correspondingly, either farfield acoustic pressure fluctuations or nearfield flow fluctuations. Emphasis is placed on the extension of resolvent analysis to stochastic systems, which allows for the treatment of wavepacket jitter, a feature known to be relevant for subsonic jet noise. Despite the simplicity of the models, they are found to qualitatively reproduce many of the features of turbulent jets observed in experiment and simulation. Sample scripts are provided and allow calculation of most of the presented results.

Efficient jet noise models using the one-way Euler equations

Experimental and numerical investigations have correlated large-scale coherent structures in turbulent jets with acoustic radiation to downstream angles, where sound is most intense. These structures can be modeled as linear instability modes of the turbulent mean flow. The parabolized stability equations have been successfully used to estimate the near-field evolution of these modes, but are unable to properly capture the acoustic field. We have recently developed an efficient method for calculating these linear modes that properly captures the acoustic field. The linearized Euler equations are modified such that all upstream propagating acoustic modes are removed from the operator. The resulting equations, called one-way Euler equations, can be stably and efficiently solved in the frequency domain as a spatial initial value problem in which initial perturbations are specified at the flow inlet and propagated downstream by integrating the equations. We demonstrate the accuracy and efficiency of the metho...

Effects of Noncircular Inlet on the Flow Structures in Turbulent Jets

Turbulent jet flows with noncircular nozzle inlet are investigated by using a Reynolds Stress Model. In order to analyze the effects of noncircular inlet, the cross section of inlet are selected as circular, square, and equilateral triangular shape. The jet half-width, vorticity thickness, and developments of the secondary flow are presented. From the result, it is confirmed that the secondary flows of square and equilateral triangular nozzle are more vigorous than that of the circular jet. This development of secondary flows is closely related to the variations of vortical motions in axial and azimuthal directions.

Correlative analysis of organised structures in turbulent jets

An approach to characterise jets by analysing the locations of large-scale instantaneous structures is presented. Planar imaging is used to identify instantaneous large-scale structures in flow fields. “Correlation Images” are generated from the auto-correlation of identified large-scale structures in instantaneous planar images. A “Structure Correlation Survey” is produced by the sum of Correlation Images from an ensemble. A Structure Correlation Survey provides a measure of the underlying large-scale structures, namely the characteristic distances and angles between large-scale structures, number densities of large-scale structures in the image field and their dominant modes of flow. The approach is assessed analytically and applied to experimental data. Four generic flow patterns are identified and used individually, or in combination, to classify jet flows. Results show that the proposed method can be used successfully to characterise jet flows based on large-scale structures in an instantaneous flow field.

Parabolized stability equation models of large-scale jet mixing noise

models to predict the evolution of low frequencies, large-scale wavepacket structures in turbulent jets and their radiated sound. We consider computations and data corresponding to high subsonic and supersonic jets from circular nozzles. Previous methods are extended to consider nonlinear interactions amongst the waves and use a Kirchhoff-surface type approach to project the near-field wavepacket amplitudes to the far-field. Linear PSE, whose initial conditions are chosen to provide an overall amplitude reference, show excellent agreement for the wavepacket amplitudes and phases with microphone array data just outside the jet shear layers, especially when the microphone data are processed to filter out contributions from uncorrelated fluctuations. Far-field sound predictions based on the linear PSE are also in reasonable agreement with far-field data. In order to investigate nonlinearity, we use an LES database to evaluate initial conditions for the PSE modes, and then compare their later evolution along the jet. Preliminary cases show some sensitivity to the initial amplitudes and their phases, and that nonlinear effects may be important in predicting the far-field sound based on the initial (near-nozzle) spectrum of disturbances.

Reprint of: Parabolized stability equation models of large-scale jet mixing noise

We report on the development of parabolized stability equation models to predict the evolution of low frequencies, large-scale wavepacket structures in turbulent jets and their radiated sound. We consider computations and data corresponding to high subsonic and supersonic jets from circular nozzles. Previous methods are extended to consider nonlinear interactions amongst the waves and use a Kirchhoff-surface type approach to project the near-field wavepacket amplitudes to the far-field. Linear PSE, whose initial conditions are chosen to provide an overall amplitude reference, show excellent agreement for the wavepacket amplitudes and phases with microphone array data just outside the jet shear layers, especially when the microphone data are processed to filter out contributions from uncorrelated fluctuations. Far-field sound predictions based on the linear PSE are also in reasonable agreement with far-field data. In order to investigate nonlinearity, we use an LES database to evaluate initial conditions for the PSE modes, and then compare their later evolution along the jet. Preliminary cases show some sensitivity to the initial amplitudes and their phases, and that nonlinear effects may be important in predicting the far-field sound based on the initial (near-nozzle) spectrum of disturbances.

Large-eddy simulations and vortex structures of turbulent jets in crossflow

Using the method of large-eddy simulation, the 3-dimensional turbulent jets in crossflow with stream-wise and transverse arrangements of nozzle are simulated, emphasizing on the dynamical process of generation and evolution of vortex structures in these flows. The results show that the basic vortex structures in literatures, such as the counter-rotating vortex pair, leading-edge vortices, lee-side vortices, hanging vortices, kidney vortices and anti-kidney vortices, are not independent physical substances, but local structures of the basic vortex structure of turbulent jets in crossflow—the 3-D stretching vortex rings originating from the orifice of the nozzle, which is discovered in this study. Therefore, the most important large-scale structures of turbulent jets in crossflow are unified to the 3-D vortex rings which stretch and twist in stream-wise and swing in transverse directions. We also found that the shedding frequencies of vortex rings are much lower than the one corresponding to the appearance of leading-edge and lee-side vortices in the turbulent jets.

Coherent Structures in Subsonic Jets: A Quasi-Irrotational Source Mechanism?

The true global source of the farfield sound radiated from a subsonic jet is the entire dynamic found within the confines of the hydrodynamic field, a dynamic which comprises an unsteady compressive excitation of the medium resulting from turbulent mixing and unsteady temperature fluctuations, these being driven by a wide range of turbulence scales. Improved understanding, and subsequent modelling of the space-time character of the global source term has been acheived in the past through study of the different physical mechanisms implicated in its dynamic. The phenomena which have to date been accepted as important in terms of the radiated sound are (i) turbulent mixing and shear, (ii) fluctuating entropy, (iii) convective amplification and (iv) refraction and scattering of sound by the mean and turbulent components of the velocity field. Once identified as important, specific modelling strategies can and have been developed in order to deal with these phenomena. The existence of coherent structures in turbulent jets was identified as important in the 1970's and their role in the production of sound has received considerable attention in more recent years. However, direct identification of the causal link between this component of the turbulence and its sound field is a delicate matter due to the virtual impossibility of directly measuring the source dynamic of the flow, this being due to an overwhelming dominance of hydrodynamic energy in the source region, and a total absence of any hydrodynamic signature in the linear, acoustic region. As a result, modelling strategies where this component of the source term is concerned have remained, at best, highly empirical. An interesting region of the flow where our understanding of the relationship between the hydrodynamic cause and its acoustic effect is concerned is the near pressure field, found just outside the rotational region of the flow. In this region the signature of the sound production mechanism and its resultant sound field are both present, coincident in space and in time. This means that pressure measurements can here be used to study the space-time structure of the two with a view to discerning the causal relationship which links them. In this work, measurements performed in the near pressure field of an isothermal subsonic jet reveal a strong interaction mechanism between the reactive and propagating components of the pressure field associated with the coherent structures of the flow, the essential features of which are captured by a simple model. On account of the success of the model it is possible to draw a number of interesting conclusions concerning the sound production mechanism associated with these structures, the most important of these being that despite the rotational velocity field in which they exist, where their sound production is concerned they can be considered as a quasi-irrotational, or wavy-wall type source mechanism. The same observation in supersonic jets has led to considerable simplifications where noise prediction is concerned – instability wave models being thus appropriate. The results from this work illustrate how similar models may be appropriate for modelling the noise production by this component of the source mechanism, believed to be predominant where peak radiation is concerned.

Coherent structures in combustion

A critical review is presented of the advances which have been made in the understanding of coherent-structure development in separated flows and of the implications of these discoveries, theoretical and practical, in the field of combustion. The main emphasis is placed on continuous combustion processes although some of the more general issues discussed relate also to transient situations. Consideration is given first to the purely fluid-mechanical aspects of coherent-structure development, this part of the review covering the findings of linear stability theory, the non-linear stages of instability growth, the development of three-dimensionality and the mixing transition in mixing layers and large structures in turbulent jets. Particular attention is given to questions of mixing and of the implications of coherent structuring for the modelling of reacting flows. Consideration is then given to the various aspects of flame behaviour which have been shown experimentally to be linked to the presence of coherent structures. Particular attention is given here to the ways in which the coherent-structure formation, and thus also the structure and behaviour of the flame, is influenced by the design of the burner equipment. Some of the areas requiring further work are identified and an extensive bibliography is provided.

Interaction of coaxial vortex rings in an ideal fluid

Problems of the motion of thin circular vortex rings in an ideal fluid are amon E the most interesting problems of vorticity dynamics. Since Helmholtz's fundamental study [i] was first published, steady interest has been generated not only by the intrinsic beauty of the problems but also by the fact that the results obtained by solving them can be used directly to explain the nature of various physical phenomena. This is illustrated both by the attempt made by Thomson (Lord Kelvin) at the end of the last century to create a vortex model of the atom [2] and by contemporary studies of the coherent structures in turbulent jets, superfluids and the wakes left by flying birds [3-5].

A study of the interactions between large-scale coherent structures and fine-grained turbulence in a round jet

Our recent studies of the interactions between large-scale structures and fine-grained turbulence in plane mixing layers have shown that many physical features of the problem may easily be obtained from an approximate energy integral description (Liu & Merkine 1976; Alper & Liu 1978), and this is confirmed by computational models (Gatski & Liu 1980). In this work, therefore, the approximate description is used to study at length the development of large-scale coherent structures in the technologically important problem of the round turbulent jet. The analysis begins from the radially integrated form of the kinetic energy equations of the mean flow, the large scale structure and the fine-grained turbulence, which are obtained through the use of the usual Reynolds time average and a conditional average with reference to the frequency of the idealized monochromatic component of the large-scale wavelike structure. This forms the basis for obtaining the ‘amplitude equations’ for the three components of the flow in terms of the mean flow momentum thickness, the large-scale structure kinetic energy and the fine-grained turbulence kinetic energy across the jet. These are obtained via the accompanying shape assumptions which also implicitly address the closure problems. The large-scale structure is also characterized by the Strouhal number St = fd/U e , where f is the frequency, d is the jet diameter and U e is the jet exit velocity, and by the azimuthal wave number n . The calculations are compared with the well controlled forced-jet observations of Binder & Favre-Marinet (1973), Favre-Marinet (1975), Favre-Marinet & Binder (1979) and Moore (1977). Although the present approximate considerations are not directed at structural details, the comparisons with observations on this aspect are most encouraging. Further theoretical work is presented that addresses the fundamental understanding of the mechanisms leading to the development of large-scale structures in turbulent jets. In general, large-scale structures in the range 0.02 ^ St < 1.60 are found first to amplify in the streamwise direction and subsequently to decay. As St is increased, the streamwise location of the peak signal moves upstream, and the streamwise lifespan shortens. Consequently, high-frequency components of the large-scale structure dominate upstream while low-frequency components prevail further downstream. The Strouhal number that gives rise to maximum amplification is about 0.70 for weak initial levels of the large-scale structure and decreases to about 0.35 for very strong initial levels. As the initial energy level of the large-scale structure increases, its maximum relative amplification decreases until a level is reached beyond which the large-scale structure decays immediately downstream. This is explained in terms of the modification of the mean flow by the increasingly high energy levels of the large-scale structure in such a manner that it chokes off its own energy supply from the mean flow. At low Strouhal numbers the n = 1 helical component amplifies initially more than the n = 0 axisymmetric component for the same initial energy level. However, the n = 1 mode decays subsequently much faster than the n = 0 mode for all Strouhal numbers. This is attributable to the azimuthally-related wave-induced turbulent shear stresses in the n = 1 mode which give rise to additional mechanisms for energy transfer to the fine-grained turbulence. The possible control of the large-scale structure is fully explored through considering adjustments of the energy levels of the fine-grained turbulence and changes in initial mean velocity profile through changes in the momentum thickness at the nozzle exit. Increasing the initial turbulence levels and smoothing the mean nozzle exit velocity profile places restraints upon the downstream amplification of the large-scale structure. The non-equilibrium development of the large-scale structure is sensitive to its own initial conditions and spectral content, the initial condition of the fine-grained turbulence and the mean flow. Physical and quantitative studies of the large-scale structure in turbulent shear flows thus necessitate that the nature of such an initial environment be established and understood.

The role of large scale structures in turbulent jets

The presence of large scale structures in circular jets is well documented in the literature over a wide range of Reynolds numbers. This presentation explores in more detail the generation and interaction and the role they play in the development of the flow field. It is based on experiments carried out, both in water and air jets, at Reynolds numbers up to 100 000.1,2 It is found, as reported by others, that the frequency of generation, as well as the various modes of the large scale vortices, can be predicted by linear stability theory. However, their further development as they are convected downstream is most difficult to study in detail. Conventional correlation measurements or sampling techniques have proven to be inadequate. In fact, the purpose for doing parallel water and jet experiments is to enable one to develop an adequate experimental method that could provide better information on the interaction process between adjacent structures. In a qualitative way, the measurements so far suggest that the induced flow field of the structures must play an important role in the interaction process and in the flow development. In the region of the potential core there is apparently an upstream influence due to the induced pressure field. This affects and, perhaps, controls such flow characteristics as the length of the potential core and the passage frequency of the structures at the end of the core. By introducing artificial disturbances into the flow further work is being carried out to clarify the above conjecture. This research was supported by the National Aeronautics and Space Administration under Grant NASA-NSG-1221.

Structure of Turbulent Jets and Wakes

The structure of two-dimensional turbulent jets and wakes is studied using two eddy viscosity models. The turbulent energy equation is used together with the mean momentum equations, and the system is closed by introducing eddy coefficients. The fine structure turbulence is obtained by applying proper boundary conditions at the mean turbulent interface position. The results, after a second averaging process, are compared with available experimental data.

Structure of turbulence

Studies of turbulent flow structure at Southampton over the past ten years have been primarily concerned with the quantitative description of those components concerned with aerodynamic noise generation and those concerned with the unsteady loading of structures. The aims of, and experimental approach to, individual experiments has changed over this period with the growth of knowledge and development of facilities. For example, excellent analogue facilities for the measurement of the statistical properties of turbulent flow structure have existed throughout the ten years while fast digital data processing equipment has recently provided what amounts to a new environment for our research. Initially the studies concerned statistical descriptions of the structure of turbulent jets and boundary layers based on time delayed cross-correlation measurements. These provide useful descriptions of practical existing flows but offer no obvious hints how these observed flows may be controlled or modified. More recently the aim has been to describe in more detail the physical structure of the individual eddies that constitute the turbulence and to clearly identify those features of the flow that led to the observed state. Three lines of active development are described, with some promising initial results. The first concerns time domain analysis of the velocity, displacement and pressure records from turbulent flows to identify the details of the characteristic eddy patterns in the flow. The second concerns potential flow modelling of impulsively started flows where arrays of elementary vortices represent vortex sheets and regions of concentrated vorticity. The third concerns modified Orr-Sommerfeld stability analysis of shear flow turbulence. These approaches are clearly complementary and so provide the basis of a comprehensive attack on the problem.