J. T. C. Liu

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/Ue, where f is the frequency, d is the jet diameter and Ue 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.

Sound generated aerodynamically revisited: large-scale structures in a turbulent jet as a source of sound

Lighthill’s formulation of the aerodynamic sound problem (Lighthill, Proc. R. Soc. Lond. A 211, 564 (1952)) is here considered as fundamental to the sound generated by real turbulent jets. For convenience, the aerodynamic sound integral is recast, via Michalke & Fuchs [J.Fluid Mech. 70, 179 (1975)), into a form involving the pressure fluctuations. It is first conjectured that the large-scale coherent structures in the turbulent jet, whose existence is now well recognized, would be responsible for the spectrally dependent highly oriented radiation patterns in the aerodynamic sound field. Accordingly, only contributions that arise from the coherent structures are retained in the aerodynamic sound integral. The neglected fine-grained turbulence as far as the sound field is concerned is thought otherwise to contribute to the broadband, nearly isotropic radiation. The present source description follows Mankbadi & Liu [Phil.Trans. R. Soc. Lond. A 298, 541 (1981)), but suitably modified to include an ensemble of n = 0 axisymmetric and n = 1 spiral modes in the relevant Strouhal number range. The coherent structures interact with the mean flow and the finegrained turbulence as an ensemble through energy exchanges dictated by rates according to their individual spectral characteristics. Because such coherent structures are relatively ‘weak’ in a real, developing turbulent jet, their mutual interactions are neglected as a first approximation. The sound sources, in a stationary coordinate system and evaluated at the appropriate retarded time, give rise to an equivalent streamwise distribution of line radiators after performance of the azimuthal and radial integrations in the aerodynamic sound integral. The streamwise oscillation of the equivalent sources is determined by an axial interference function strongly influenced by the wavenumber of each individual mode whereas the streamwise growth and decay of the source envelope is determined primarily by the coherent structure amplitude whose spectral dependence is also strong. The streamwise net imbalance of the source contribution, reflected by the axial integration in the aerodynamic sound integral, gives rise to the far sound field. It is found that in general, the radiation is primarily in the direction of the jet exhaust; the radiation patterns of the n = 0 modes resembling those of longitudinal quadrupoles and those of the n = 1 modes resembling those of lateral quadrupoles. However, the n = 0 modes tend to peak at Strouhal numbers less than those of the n = 1 modes. The superposition gives a directional-spectral behaviour that strikingly resembles that of observations: lower frequency sound radiates preferentially in the forward direction and as the frequency increases, the peak radiation moves towards the lateral directions; it is also found that contributions to the high-frequency sound come from coherent structures that peak nearer the nozzle lip, whereas contributions to the low-frequency sound come from such structures that peak further downstream in the jet. The calculated spectral shapes are narrower than observations by typically a deficit of 4—7 dB per octave on both the high and low frequency sides and this is most likely attributable to the nearly isotropic radiation caused by the broad-band fine-grained turbulence whose direct contribution to the sound field is not accounted for. For the same reason, the calculated aerodynamic sound field has a large deficit compared with observations in the vicinity of the 90- degree region. The dominant contributions to the radiation come from the so-called shear noise in the forward arc, whereas both the shear and self-noise of the coherent structures become equally insignificant to the same order in the 90-degree region. Although the source distribution within the jet is calculated for an identically incompressible fluid, it is used in a limited sense to study the effect of jet exit velocity on the peak radiation frequency in the forward direction: it is found that the peak value of fd/a0, where f is the frequency, d the jet nozzle diameter and a0 the ambient sound speed, take on a value of about 0.30 independently of the jet velocity and this compares favourably with an observational value of about 0.20. In general, the angular distribution of the peak frequency due to coherent structures radiation compared favourably with observations. Compressibility effects that somewhat limit the amplification of coherent structures, as well as the effects of higher azimuthal modes whose radiation would peak at higher frequencies and larger lateral directions, warrant further study in the light of the present considerations. The present work, however, has already shown that the consequences of Lighthills formulation of the aerodynamic sound problem agree with major features of observations and that this is brought about by taking into account as sources the growing and decaying largescale coherent eddies whose development within the turbulent jet and whose radiational properties are all strongly dependent upon their spectral contents.

On the growth of mushroomlike structures in nonlinear spatially developing Goertler vortex flow

Spatially developing longitudinal vortices, originating from upstream Goertler vortices, are studied by a finite‐difference algorithm. Three‐dimensional parabolized Navier–Stokes equations for a ‘‘slightly’’ concave wall and a thin boundary layer are used, in conjunction with the total (mean plus perturbation) flow quantities as dependent variables. The behavior of Goertler vortex development from the linear region to the fully nonlinear region was obtained just prior to the development of secondary instabilities and turbulence. Comparisons with the experiments are discussed, showing the development of mushroomlike features in the streamwise velocity component.

On the mechanism of sinuous and varicose modes in three‐dimensional viscous secondary instability of nonlinear Görtler rolls

The present study begins with the solution of the linearized partial differential equations for viscous, secondary instabilities of nonlinearly developed primary Gortler rolls. A global energy balance verifies the amplification rates obtained from secondary instability analysis and confirms that the sinuous mode would dominate. This therefore explains the strong resemblance between the rms‐streamwise u‐velocity structure obtained by hot‐wire measurements [Swearingen and Blackwelder, J. Fluid Mech. 182, 255 (1987)] in the cross‐sectional yz plane and that of the sinuous mode. The u contours of the secondary disturbance have peaks at locations corresponding to the shoulder regions of the mushroom‐like primary streamwise U‐velocity structure and even more intense regions surrounding the mushroom stem near the wall. This structure is explained by detailed analysis of the energy‐balancing mechanisms for u2/2 in yz plane: the Reynolds stress‐conversion mechanism predominantly affects the primary spanwise rate o...

The secondary instability in Goertler flow

The ‘‘high‐frequency’’ secondary instability of nonlinearity developed and strongly modified Goertler vortices is numerically studied by considering two kinds of secondary instability modes, i.e., sinuous modes and varicose modes. A spectral collocation method with Chebyshev polynomials is used for calculations. It is found that secondary instabilities are more correlated with the vertical vorticity and the sinuous type disturbance would prevail over the varicose one for the same frequency and streamwise wave number found in experiments.

Note on a Wave-Hierarchy Interpretation of Fluidized Bed Instabilities

The technological importance of fluidized beds is well known. Experimental observations indicate that a uniformly fluidized bed is unstable to small planar voidage disturbances that grow exponentially upwards along the bed and, under certain circumstances, attain saturation at finite amplitudes. The linearized theory is recast into a form that shares mathematical similarities and interpretations with ‘wave hierarchies’. In wave hierarchies, the higher-order waves, which correspond to higher-frequency waves nearer the distributor, have wave speeds that are lower than those of the lower-order waves, and the stability condition is thus violated. The lower-order waves correspond to these lower-frequency waves further along the fluidized bed. Thus voidage signals grow exponentially along the higher-order waves. Such waves will eventually be left behind and surpassed by the faster lower-order waves along which voidage perturbations will be ‘focused’ owing to an effective negative diffusion coefficient. The peculiar double peaks in voidage perturbation waves observed by El-Kaissy and Homsy appear to be explicable by the present wave-hierarchy interpretation of the linearized theory.

Nonlinear binary-mode interactions in a developing mixing layer

Formulation et resultats des interactions a deux ondes dans une couche de cisaillement spatialement developpee

Flow Induced by an Oscillating Infinite Flat Plate in a Dusty Gas

The flow induced in an incompressible dusty gas by an infinite flat plate oscillating in its own plane is studied. The differential equation describing this problem is given; its form exhibits the relaxation from a frozen‐diffusion process (corresponding to a clean gas) to an equilibrium‐diffusion process (corresponding to a single heavier gas). The gas velocity profile, shear stress on the plate, and the particle velocity profile are obtained exactly and are discussed in terms of the parameters of the problem. The essential feature is the inhibition of viscous diffusion in the gas by the particle‐gas velocity relaxation. Mechanical energy dissipation is discussed.

Nonlinear Development of an Instability Wave in a Turbulent Wake

Recent experimental observations appear to indicate that turbulent mean flow velocity profiles with points of inflection are dynamically unstable with respect to traveling wavy disturbances. As an illustration, the description of the nonlinear development of an instability wave in the turbulent wake behind a thin body is presented on the basis of integrals of the equations of mean flow momentum, kinetic energy, and the time‐averaged disturbance kinetic energy, with the turbulent dissipation integrals relegated to phenomenological considerations. The over‐all physical mechanisms leading to the streamwise distribution of the mean flow decay and disturbance energy amplification and decay are explained through the disturbance energy production integral and the mean flow and disturbance turbulent‐dissipation integrals. The growth of the disturbance wave is limited by (1) the rapidity of turbulent mean flow decay which renders the production integral less spectacular and (2) the prominant role of the turbulent ...

On the development of noise-producing large-scale wavelike eddies in a plane turbulent jet

In this paper we study the development of large-scale wavelike eddies in a two-dimensional turbulent jet, extending earlier work on the mixing region (Liu 1974). The basic mean flow develops from one of mixing-region type with an initially specified boundary-layer thickness into a fully developed jet. This study brings out the role of the varicose and sinuous modes as they develop in a growing mean flow. In general, it is found that, for a given frequency parameter, the varicose mode has a shorter streamwise lifetime than the sinuous mode. For lower frequencies, the latter persists past the end of the potential core only to become subject to dissipation by the enhanced fine-scale turbulent activity in that region.