Developing Shear Flow Research Articles

Reacting and non-reacting, three-dimensional shear layers with spanwise stretching

A three-dimensional, steady, laminar shear-layer flow spatially developing under a boundary-layer approximation with mixing, chemical reaction, and imposed normal strain is analyzed. The purpose of this study is to determine conditions by which certain stretched vortex layers appearing in turbulent combustion are the asymptotic result of a spatially developing shear flow with imposed compressive strain. The imposed strain creates a counterflow that stretches the vorticity in the spanwise direction. Equations are reduced to a two-dimensional form for three velocity components. The non-reactive and reactive cases of the two-dimensional form of the governing equations are solved numerically, with consideration of several parameter inputs, such as the Damköhler number, the Prandtl number, chemical composition, and free-stream velocity ratios. The analysis of the non-reactive case focuses on the mixing between hotter gaseous oxygen and cooler gaseous propane. The free-stream strain rate κ* is predicted by ordinary differential equations based on the imposed spanwise pressure variation. One-step chemical kinetics are used to describe diffusion flames and multi-flame structures. The imposed normal strain rate has a significant effect on the width of downstream mixing layers as well as the burning rate. Asymptotically in the downstream direction, a constant width of the shear layer is obtained if the imposed normal strain rate is constant. The one-dimensional asymptotic result is an exact solution to the multicomponent Navier–Stokes equation for both reacting and non-reacting flows, although it was obtained using the boundary-layer approximation. A similar solution with the layer width growing with the square root of downstream distance is found when the imposed strain rate decreases as the reciprocal of downstream distance. The reduced-order asymptotic solutions can provide useful guidance in developing flamelet models for simulations of turbulent combustion.

The Development and Application of a Kinetic Theory for Modeling Dispersed Particle Flows

Abstract This Freeman Scholar article reviews the formulation and application of a kinetic theory for modeling the transport and dispersion of small particles in turbulent gas-flows. The theory has been developed and refined by numerous authors and now forms a rational basis for modeling complex particle laden flows. The formalism and methodology of this approach are discussed and the choice of closure of the kinetic equations involved ensures realizability and well posedness with exact closure for Gaussian carrier flow fields. The historical development is presented and how single-particle kinetic equations resolve the problem of closure of the transport equations for particle mass, momentum, and kinetic energy/stress (the so-called continuum equations) and the treatment of the dispersed phase as a fluid. The mass fluxes associated with the turbulent aerodynamic driving forces and interfacial stresses are shown to be both dispersive and convective in inhomogeneous turbulence with implications for the build-up of particles concentration in near wall turbulent boundary layers and particle pair concentration at small separation. It is shown how this approach deals with the natural wall boundary conditions for a flowing particle suspension and examples are given of partially absorbing surfaces with particle scattering and gravitational settling; how this approach has revealed the existence of contra gradient diffusion in a developing shear flow and the influence of the turbulence on gravitational settling (the Maxey effect). Particular consideration is given to the general problem of particle transport and deposition in turbulent boundary layers including particle resuspension. Finally, the application of a particle pair formulation for both monodisperse and bidisperse particle flows is reviewed where the differences between the two are compared through the influence of collisions on the particle continuum equations and the particle collision kernel for the clustering of particles and the degree of random uncorrelated motion (RUM) at the small scales of the turbulence. The inclusion of bidisperse particle suspensions implies the application to polydisperse flows and the evolution of particle size distribution.

Space-time correlations of velocity in a Mach 0.9 turbulent round jet

In a turbulent jet, the numerical investigation of space-time correlations C(r, τ) at two-point and two-time of streamwise fluctuating velocities is presented along the nozzle lipline. Large-eddy simulation (LES) is performed for a Mach 0.9 turbulent jet issuing from a round nozzle. The turbulent boundary layer is well developed at the nozzle outlet, upon the inner wall, by adopting synthetic turbulent inlet boundary conditions. We study the cross correlations of streamwise fluctuating velocities at three particular streamwise positions, i.e., x = 0.71, 7.03, and 34.47r0, corresponding to different stages of jet development, where r0 is the radius of the nozzle. Present results show that the classical Taylor’s frozen-flow model is unable to predict C(r, τ) accurately in this strongly spatially developing shear flow since the distortion of the flow pattern is missing. The isocorrelation contours of C(r, τ) show a clearly elliptical feature, which is found to be well predicted by the elliptic approximation (EA) model [G.-W. He and J.-B. Zhang, “Elliptic model for space-time correlations in turbulent shear flows,” Phys. Rev. E 73, 055303 (2006)]. According to the EA model, C(r, τ) has a scaling form of C(rE, 0) with two characteristic velocities U and V, i.e., rE = (r − Uτ)2 + V2τ2. By examining LES data, it is found that the characteristic velocity U determined in LES is in general consistent with the theoretical Ut in the EA model, while the trend of V in LES also matches with that of the theoretical Vt. Additionally, it is interesting that the ratio of V to Vt is approximately a constant V/Vt ≃ 1.3 in the turbulent jet.

Nonlinear Theories for Shear Flow Instabilities: Physical Insights and Practical Implications

This article reviews the nonlinear stability theories that have been developed to explain laminar–turbulent transition processes in boundary and free shear layers. For such spatially developing shear flows, a high–Reynolds number approach is necessary to account for, in a systematic and self-consistent manner, multiple competing physical factors, such as nonlinearity, nonparallelism, nonequilibrium, and viscosity. While the basic ideas and fundamental mechanisms are rooted in the classical weakly nonlinear theory, which was formulated primarily for exactly parallel flows and on the basis of finite Reynolds number, the high–Reynolds number formulations lead to low-dimensional evolution systems, which differ significantly from the finite–Reynolds number counterparts and better describe the observations. Owing to efforts in the past 30 years or so, nonlinear evolution systems have been derived for inviscid Rayleigh modes, viscous Tollmien–Schlichting waves, (first and second) Mack modes, and cross-flow vortices. Theories have also been developed for nonlinear intermodal interactions, including oblique mode interaction, subharmonic resonance, phase-locked interactions, and fundamental resonance; these underpin many intriguing behaviors in the three-dimensional stages of transition. These theories and results explain several key nonlinear features observed and should play a useful role in guiding future experimental and numerical investigations.

Three-Dimensional Aerators: Characteristics of the Air Bubbles

Three-dimensional aerators are often used in hydraulic structures to prevent cavitation damage via enhanced air entrainment. However, the mechanisms of aeration and bubble dispersion along the developing shear flow region on such aerators remain unclear. A double-tip conductivity probe is employed in present experimental study to investigate the air concentration, bubble count rate, and bubble size downstream of a three-dimensional aerator involving various approach-flow features and geometric parameters. The results show that the cross-sectional distribution of the air bubble frequency is in accordance with the Gaussian distribution, and the relationship between the air concentration and bubble frequency obeys a quasi-parabolic law. The air bubble frequency reaches an apex at an air concentration (C) of approximately 50% and decreases to zero as C = 0% and C = 100%. The relative location of the air-bubble frequency apex is 0.210, 0.326 and 0.283 times the thickness of the layers at the upper, lower and side nappes, respectively. The air bubble chord length decreases gradually from the air water interface to the core area. The air concentration increases exponentially with the bubble chord length. The air bubble frequency distributions can be fit well using a “modified” gamma distribution function.

Yield stress and onset of nonlinear time-dependent rheological behaviour of gellan fluid gels

We present a method to determine simultaneously the yield stress and the onset of non-linear time-dependent rheological behaviour applied to gellan gum fluid gels (0.2wt% gellan gum, 0.2wt% Ca+2 as promoters of the formation the gel). It consists of creep-recovery-creep tests for shear stresses in the range 2–6Pa, avoiding the effect of shear history. Results obtained at the lower stresses applied showed a response dominated by viscoelasticity. At a given higher stress the variability of results obtained allowed the yield stress to be determined, which is associated to the occurrence of shear banding. Above this practical yield stress the results showed a clear thixotropic response. A flow curve free of shear history effects could be derived from series of creep experiments and showed very shear thinning behaviour. The shear rate dependence of viscosity within the fully developed shear flow region fitted the power law equation with a flow index of 0.2 and the comparison of viscosities calculated at different shear times illustrated the thixotropic behaviour of gellan fluid gels.

Stability of Disturbance Waves in Developing Shear Flows: A Review of ‘Ad-hoc’ Methods

This paper attempts to induce some formalism in the study of stability of developing shear flows, by use of so called ‘ad-hoc’ methods wherein all terms upto and inclusive of a particular order are lumped together in the same equation. However, whether the ‘ordering’ is perturbation or asympotic is somewhat vague. The classical example of stability of a developing shear flow is the stability problem of boundary-layer flow over a flat-plate including the non-parallel effects. Other examples are free shear flows; also, problems involving flow over alternate rigid and porous panels with suction, or suction from one wall and injection from the opposite wall. The quick method for stability analysis is to use the local Orr-Sommerfeld solution, using the quasi-parallel approximation. This is sometimes augmented by inclusion of some non-parallel terms, perhaps in an ad-hoc manner, and so called ‘improved solutions’ are obtained. The present paper attempts to establish a formal framework for these methods so that these methods are no more termed ‘ad-hoc’.

Intermittent Bursts in the Boundary Plasma of HT-7

Intermittent characteristics of turbulence induced by coherent structures (blobs) was clearly observed in the ion saturation current signal, density fluctuation, particle flux and heat flux in HT-7. It is obvious that ion saturation current signal has deviated from Gaussian distribution and the skewness (S) and flatness (K) of signal increase radially outwards in the scrape-off layer (SOL). Using conditional analysis (CA), asymmetric character of the intermittent bursts are demonstrated. Owing to the radial propagation of the coherent structures, the particle density profile in SOL is non-exponential and flat outwards from the last close flux surface (LCFS). It is found around LCFS that the large burst fluctuations (above 2.5 rms) are responsible for about 50% of the total particle transport. Burst events move radially outwards with Eθ×B velocity, and the blob size can be calculated as δr ∼ Vrτc. Our experiment shows that the blob size, life time and drift velocity experienced a pronounced decorrelation in the shear layer. The electrostatic Reynolds stress components become very strong and show a radially steep gradient in the proximity of the shear layer. These experimental findings may imply that the coherent structures are titled by the developed shear flow in the E × B shear layer.

Coulomb–Mohr Granular Materials: Quasi-static Flows and the Highly Frictional Limit

One approach to modeling fully developed shear flow of frictional granular materials is to use a yield condition and a flow rule, in an analogous way to that commonly employed in the fields of metal plasticity and soil mechanics. Typically, the yield condition of choice for granular materials is the Coulomb–Mohr criterion, as this constraint is relatively simple to apply but at the same time is also known to predict stresses that are in good agreement with experimental observations. On the other hand, there is no strong agreement within the engineering and applied mechanics community as to which flow rule is most appropriate, and this subject is still very much open to debate. This paper provides a review of the governing equations used to describe the flow of granular materials subject to the Coulomb–Mohr yield condition, concentrating on the coaxial and double-shearing flow rules in both plane strain and axially symmetric geometries. Emphasis is given to highly frictional materials, which are defined as those granular materials that possess angles of internal friction whose trigonometric sine is close in value to unity. Furthermore, a discussion is provided on the practical problems of determining the stress and velocity distributions in a gravity flow hopper, as well as the stress fields beneath a standing stockpile and within a stable rat-hole.

Acoustic near field of a transonic instability wave packet

We consider the problem of acoustic radiation generated by a spatial instability wave on a weakly developing shear flow. Assuming a local WKBJ approximation for the instability wave near its maximum, we compute the acoustic pressure field by using a Fourier transform along the streamwise direction. When the instability wave is close to transonic near its maximum amplitude, approximations for this pressure field are obtained by a steepest descent method. A branch cut and several saddle points are shown possibly to contribute to the approximation. A detailed analysis of these contributions is provided. The modifications of the acoustic field when we pass from subsonic to supersonic are examined. In particular, the superdirective character of the acoustic field of subsonic instability waves and the directivity pattern of supersonic waves are shown to be both compatible with our mathematical description and associated with a single saddle-point contribution.The acoustic near field is also shown to possess a caustic around which a specific approximation is derived. In a large region of the physical space, the near field is composed of two saddle-point contributions. Close to the shear flow, one of these contributions degenerates into a branch-point contribution which always becomes dominant over the instability wave downstream of a location that is computed. An interesting phenomenon is observed in certain regions downstream of the maximum: the transverse behaviour of the instability wave has to be exponentially growing far from the shear layer to match the acoustic field. We demonstrate that this phenomenon neither requires a branch-point contribution nor a supersonic instability wave.

Bubbly flow structure in hydraulic jump

In an open channel, a hydraulic jump is the rapid transition from super- to sub-critical flow associated with strong turbulence and air bubble entrainment in the mixing layer. New experiments were performed at relatively large Reynolds numbers using phase-detection probes. Some new signal analysis provided characteristic air–water time and length scales of the vortical structures advecting the air bubbles in the developing shear flow. An analysis of the longitudinal air–water flow structure suggested little bubble clustering in the mixing layer, although an interparticle arrival time analysis showed some preferential bubble clustering for small bubbles with chord times below 3 ms. Correlation analyses yielded longitudinal air–water time scales T x x V 1 / d 1 of about 0.8 in average. The transverse integral length scale Z / d 1 of the eddies advecting entrained bubbles was typically between 0.25 and 0.4, irrespective of the inflow conditions within the range of the investigations. Overall the findings highlighted the complicated nature of the air–water flow.

New Physically Based Approach of Mass Conservation Correction in Level Set Formulation for Incompressible Two-Phase Flows

A novel physically based mass conservation model is developed in the framework of a level set method, as an alternative to the Heaviside function based formulation classically employed in the literature. In the proposed “volume fraction based level set approach,” expressions for volume fraction function for each interfacial computational cell are developed, and are subsequently correlated with the corresponding level set functions. The volume fraction function, derived from a physical basis, is found to be mathematically analogous to the Heaviside function, except for a one-dimensional case. The results obtained are compared with the benchmark experimental and numerical results reported in the literature. Finally, transient evolution of a circular bubble in a developing shear flow and rising bubbles in a static fluid, are critically examined. The Cox angle and the deformation parameter characterizing the bubble evolution are critically examined. An excellent satisfaction of the mass conservation requirements is observed in all case studies undertaken.

Characteristics of Undular Hydraulic Jumps: Experiments and Analysis

The writers measured velocity, pressure and energy distributions, wavelengths, and wave amplitudes along undular jumps in a smooth rectangular channel 0.25 m wide. In each case the upstream flow was a fully developed shear flow. Analysis of the data shows that the jump has strong three-dimensional features and that the aspect ratio of the channel is an important parameter. Energy dissipation on the centerline is far from negligible and is largely constrained to the reach between the start of the lateral shock waves and the first wave crest of the jump, in which the boundary layer develops under a strong adverse pressure gradient. A Boussinesq-type solution of the free-surface profile, velocity, and energy and pressure distributions is developed and compared with the data. Limitations of the two-dimensional analysis are discussed.

Viscous structure of plane waves in spatially developing shear flows

This paper is concerned with the propagation of linear plane waves in incompressible, two-dimensional weakly nonparallel shear flows for large Reynolds numbers. Waves are analyzed for arbitrary complex frequency ω and local wave number k when nonparallel effects are assumed to be due to weak viscous diffusion. The inviscid approximation is shown to correctly describe, at leading order, the cross-stream variations of local plane waves at all stations where they are locally amplified in a frame of reference moving at the local phase speed ℜeω/ℜek, i.e., at stations where the temporal growth rate σ≡𝔉mω−𝔉mk ℜeω/ℜek remains positive. This result also holds as long as the local phase speed lies outside the range of values reached by the basic velocity profile. By contrast, the inviscid approximation fails to represent cross-stream variations in the critical layers when waves are locally neutral (σ=0), and in large viscous regions when they become damped (σ<0). Uniformly valid WKBJ approximations are derived in these regions and the results are applied to the description of forced spatial waves and self-excited global modes. © 1995 American Institute of Physics.

MODELING OF THE REACTANT CONVERSION RATE IN A TURBULENT SHEAR FLOW

Results are presented of direct numerical simulations (DNS) of spatially developing shear flows under the influence of infinitely fast chemical reactions of the type A + B yields Products. The simulation results are used to construct the compositional structure of the scalar field in a statistical manner. The results of this statistical analysis indicate that the use of a Beta density for the probability density function (PDF) of an appropriate Shvab-Zeldovich mixture fraction provides a very good estimate of the limiting bounds of the reactant conversion rate within the shear layer. This provides a strong justification for the implementation of this density in practical modeling of non-homogeneous turbulent reacting flows. However, the validity of the model cannot be generalized for predictions of higher order statistical quantities. A closed form analytical expression is presented for predicting the maximum rate of reactant conversion in non-homogeneous reacting turbulence.

The structure of a passive scalar field with a uniform mean gradient in rapidly sheared homogeneous turbulent flow

The effect of an arbitrarily oriented mean passive scalar gradient on one-point passive scalar statistics is studied in homogeneous turbulent shear flow in the limit of rapid shearing. By neglecting the nonlinear inertial transfer to small scales an analytical solution for individual Fourier modes is obtained for the case of unity Prandtl number. This solution is used to compute the development of one-point statistics for the velocity and scalar fields in the inviscid limit. Comparisons to direct numerical simulations of the full nonlinear equations for the same flow show that in addition to describing the early time response to the imposed shear, the linear solution gives reasonable estimates of several correlation coefficients for the developed shear flow.

Fully Developed Plastic Shear Flow of Granular Materials

Synopsis A consistent mechanical basis is presented for the mathematical description of plane, plastic shear deformation of frictional, cohesive or cohesionless granular materials. The deformation considered is fully developed flow of particles which override each other over long distances. It is an idealization of the cataclastic flow by which all faulting processes in loose or slightly consolidated sediments proceed. It is shown that granular material in fully developed shear flow exhibits all the essential properties of a perfectly plastic material. In particular, it lacks a one-to-one correspondence between strain rates and stresses, as is explained by a discussion of the energy dissipation involved. Instead the relation between stress and strain increments reduces to a relation between corresponding principal directions of the two tensors. Despite the plastic character of the flow, the flow rules obtained by employing the rheological concept of perfect plasticity to granular materials (perfect soil plasticity) are definitely inadequate for a description of fully developed shear flow. This makes it necessary to search for a further property of overriding shear flow which, when used to supplement perfect plasticity, may make that concept an adequate model. The desired property is found in the non-coaxiality of the tensors of stress and strain increment. Arguments are given, both on microand macro-levels, which support the view that in fully developed shear flow of a perfectly isotropic granular material the internal kinematic constraints, which are associated with the overriding mode of the motion, cause the principal axes of stress to deviate from the corresponding principal axes of strain rate. It is shown in extenso that this non-coaxiality is perfectly compatible with material isotropy as long as the deformational response of the material is plane. The non-coaxial flow rules are formulated and contrasted with the flow rules of perfect soil plasticity. The mathematical formalism that describes the deformation field differs only in minor details from that of the theory of perfect metal plasticity. On présente une base mécanique offrant un caractère de continuité pour la description mathématique de la déformation de cisaillement plastique, plane de matériaux granulaires en état de frottement, cohérents ou non cohérents. La déformation considérée est un écoulement de particule complètement développé, ces particules se chevauchant les unes les autres sur de longues distances. C'est une idéalisation de l'ecoulement cataclastique par suite duquel se produisent les opérations de détection de défauts dans des sédiments sans cohésion ou légèrement consolidès. On fait apparaître qu'un matériau granulaire dans un écoulement de cisaillement complètement développé fait preuve de toutes les propriétés essentielles d'un matériau parfaitement plastique. En particulier, la correspondance de un pour un entre les taux de déformation et les contraintes lui fait défaut, comme l'explique une discussion sur la dissipation d'energie entraînée. Au lieu de cela, le rapport entre les accroissements de déformation et de contrainte se réduit à un rapport entre les directions principales correspondantes des deux tenseurs. Malgré la nature plastique de l'ecoulement, les règles régissant l'ecoulement obtenues en utilisant le concept rhéologique de la plasticité parfaite dans le cas des matériaux granulaires (plasticité parfaite du sol) sont absolument insuffisantes pour une description d'un Ccoulement de cisaillement complètement développé. Cela rend nécessaire la recherche d'une propriété supplémentaire d'écoulement de cisaillement à chevauchement, qui lorsqu 'elle est utilisée pour compléter la plasticité parfaite, pourra rendre ce concept un modèle suffisant. La propriété requise se trouve dans l'état non coaxial des tenseurs d'accroissement de déformation et de contrainte. Des arguments sont fournis, aux niveaux à la fois microscopiques et macroscopiques, qui appuyent l'opinion selon laquelle dans un écoulement de cisaillement complètement développé d'un matériau granulaire parfaitement isotrope, les contraintes cinématiques internes qui s'associent au mode de mouvement à chevauchement, font dévier les principaux axes de contrainte par rapport aux principaux axes correspondant du taux de déformation. On fait apparaître de façon détaillée que cet état non coaxial est parfaitement compatible avec l'isotropie du matériau tant que la réaction du matériau du point de vue de la déformation a lieu dans le plan. On formule les règles d'écoulement non coaxial et on fait ressortir les contrastes qu'elles présentent avec les règles d'ecoulement relativement à la plasticité parfaite du sol. Le formalisme mathématique qui décrit le champ de déformation n'est différent que dans le cas de détails secondaires de celui de la théorie de la plasticité parfaite d'un métal.