On the role of unsteady forcing of tracer gradient in local stirring

Local stirring properties in two basic mixing flows – namely, the blinking vortex and the sine flow – are studied through the tracer gradient approach. The velocity gradient tensor and related quantities such as the strain persistence parameter are derived from the analytical velocity fields. Numerical Lagrangian tracking of the gradient of a tracer shows how local stirring is affected by forcing experienced through strain persistence. In both flows Lagrangian variations of strain persistence occuring on a time scale shorter than the response time scale of the tracer gradient lead the latter to align close to the direction determined by the mean strain persistence. It is the special alternating behaviour of strain persistence resulting from flow operation that makes this direction coincide with the local compressional strain direction for both the sine flow and the clockwise/counterclockwise blinking vortex. The rise of the tracer gradient and thus local stirring are in turn promoted by this statistical alignment.     

Nonstationary aspects of passive scalar gradient behaviour

The dynamics of a passive scalar gradient experiencing fluctuating velocity gradient through the Lagrangian variations of strain persistence is studied. To this end, a systematic, numerical analysis based on the equation for the orientation of the gradient of a nondiffusive scalar in two-dimensional flow is performed. When the gradient responds weakly its orientation properties are determined by the mean value of strain persistence. Statistical alignment of the scalar gradient with the direction defined by the opposed actions of strain and rotation, by contrast, requires the gradient to keep up with strain persistence fluctuations. These results have been obtained for both strain- and effective-rotation-dominated regimes and are supported by relevant experimental data. Consequences of the unsteady behaviour of the scalar gradient on mixing properties are also analysed.     

Comparison of Observations from a Laboratory Model with Stochastic Theory: Initial Analysis of Hydraulic and Tracer Experiments

Hydraulic and tracer tests were conducted in a flow cell containing a mixture of sediments designed to mimic a two-dimensional, log-normally distributed, second-order stationary, exponentially correlated random conductivity field. With 60 integral scales in the direction of mean flow and 25 integral scales perpendicular to this direction, behavior of flow and transport in the interior of the flow cell can be compared directly with stochastic solutions for flow and transport. Using 144 piezometers and 361 platinum electrodes, the distribution of hydraulic head and the concentrations of an ionic tracer could be monitored in substantial detail. The present discussion presents the details of the experimental equipment. Results and initial analysis of hydraulic measurements and characterization of a two-dimensional tracer plume are also presented. Analysis using first-order hydraulic theory shows that the flow through the medium was consistent with an effective conductivity equal to the geometric mean of the conductivity distribution. Further, the semivariogram of head increments as observed in the experimental results was consistent with the semivariogram predicted by theory. The chemical transport experiments are here compared with the early solutions presented by Dagan (1984, 1987). The observed rate of longitudinal spread of two tracer plumes was slightly less than that predicted using this theory. Further, the spread in the transverse dimension was observed to decline from the initial plume dimensions and then remain constant or increase slightly, but at a rate lower than predicted by the theory. The difference between the hydraulic and transport results is believed to be related to the fact that the hydraulic results were averaged over a very large portion of the flow cell such that ergodic conditions could be assumed. In contrast, the initial geometry of the plume covered only approximately five integral scales in the transverse direction such that the validity of the assumption of ergodic conditions must be questioned in the analysis of results for the chemical transport.     

Impact of density stratification on the global mode in a swirling jet: Stochastic modelling and Lagrangian coherent structures

In an experimental investigation, the stochastic dynamics of the global mode in a turbulent swirling jet are considered. From the application of the swirling jet in gas turbine combustors, it was observed that a specific density gradient in the flow leads to a suppression of the global mode. This phenomenon was replicated in a generic swirling jet using an electrical heating coil placed inside the breakdown bubble. In the present investigation, the dynamics of the global mode obtained from PIV and pressure measurements are analysed using a stochastic reduced-order model to describe the instability. The stochastic model is necessary to explain the interaction between the deterministic dynamics of the global mode and the perturbations by the background turbulence. The calibration of the stochastic model provides the amplification rate of the global mode that defines the transition of the flow, dependent on the swirling strength and the density difference. The spatial structure of the global mode is further investigated from Lagrangian coherent structures of the flow field which are computed from the 3D time-resolved velocity field reconstruction based on planar PIV measurements. The Lagrangian visualisations and schlieren visualisations are used to explain the absence of the density effects on the global mode at larger Reynolds numbers. The analysis gives a detailed view of the stochastic dynamics of a hydrodynamic instability in a turbulent flow.     

Modélisation de la turbulence en déclin avec rotation

The study presented in the present paper deals with the modeling of the rotation effects on a homogeneous turbulence evolving without a mean velocity gradient. The developed model results from a modification of the second order model of Haworth and Pope. This modification is expressed as a function of the rotation rate, of the component of the Reynolds tensor and of the characteristic length scale in the direction of the rotation axis. The obtained results have evidenced the great capacity of the present model to describe the rotation effects on the time evolution of the homogeneous decaying turbulence.     

Analysis of small-scale scalar mixing processes in highly under-expanded jets

In the present study, we numerically investigate turbulent scalar mixing taking place downstream of highly under-expanded jets. The focus is placed on two inter-related issues: (i) the closure of the mean scalar dissipation rate (SDR) and (ii) the turbulence–scalar interaction (TSI) term. It is indeed commonly admitted that the former, i.e., the SDR, which is defined as the product of the scalar diffusivity with the squared scalar gradient, provides a good measure of the mixing efficiency. In turbulent flows, the mean (turbulent) SDR requires a specific closure to be settled. It is generally obtained within the approximation of a linear relaxation of scalar fluctuations or linear relaxation model. We will first evaluate herein the performance of this widely used closure. The analysis is further developed by means of the consideration of the mean SDR transport equation which shows that, in gaseous conditions, the SDR is mainly driven by two terms: (i) a dissipation contribution and (ii) the third-order correlation between the velocity gradient tensor and small-scale scalar anisotropy tensor. The scalar mixing efficiency thus appears to be controlled by the latter quantity, which is often denoted as the TSI term. It can be shown that only the symmetric part (rate of strain) of the velocity gradient tensor contributes to this term; the anti-symmetric part modifies, indeed, the orientation of the scalar gradient but not its magnitude. The classical approach is to analyze this contribution in the eigenframe of the rate of strain tensor. Such analyses show that, in homogeneous isotropic turbulence, the scalar gradient tends to align with the most compressive direction, thus leading to SDR production. However, the present conditions, which are far from homogeneity and involve strong density variations, may modify this classical picture. The present study analyzes this possible influence.     

Dynamics of swirling jet flows

Experimental investigations of near-field structure of coaxial flows are presented for four different configurations: coaxial jets without rotation (reference case), outer flow rotating only (OFRO), inner-jet rotating only (IJRO) and corotating jets (CRJ). The investigations are performed in a cylindrical water tunnel, with an independent rotation of two coaxial flows. Laser tomography is used to document the flow field, and photographs are shown for different configurations. Time mean velocity profiles obtained by PIV, with and without swirl, are also presented. The dynamics of the swirling jets in the initial region (i.e. near the exit of the jets) is described. The effects of azimuthal velocity and axial velocity ratio variations on flow dynamics are examined. The appearance and growth of the first instabilities are presented and compared with some theoretical results, as is the influence of the rotation (inner or outer) on the dominating structures.     

An Experimental Study of the Rotational Effects on Separated Turbulent Flow During Stall Delay

Three-dimensional velocity fields were measured using tomographic particle image velocimetry (Tomo-PIV) on a model of the blade of a small-scale horizontal axis wind turbine (HAWT) to study the effects of rotation on separated turbulent flows during stall delay at a global tip speed ratio (TSR) of 3 and a Reynolds number of 4800. The flow fields on a static airfoil were also measured at a similar angle-of-attack (AOA) and Reynolds number for comparison. It was observed that the blade’s rotation in the streamwise direction significantly affected both the mean flow and the turbulence statistics over the suction surface. The mean velocity fields revealed that, different from the airfoil flow at large AOA, the recirculation region with reversed flow did not exist on the suction surface of the blade and the flow was rather attached. Mean spanwise flow from blade’s root to its tip was also generated by the rotation. The mean vorticity vector of the blade flow was found to be tilted in the rotational direction of the blade, as well as in the wall-normal direction. Of particular effects of the rotation on Reynolds stresses were the enhancement of 〈w ²〉 and the creation of strong 〈v w〉. The production of Reynolds stresses was also affected by blade’s rotation directly through the rotational production terms and indirectly by dramatically changing the fluctuating velocity fields. The distribution of enstrophy was observed to be modified by rotation, too.     

An interacting micro-crack damage model for failure of brittle materials under compression

A model is developed for brittle failure under compressive loading with an explicit accounting of micro-crack interactions. The model incorporates a pre-existing flaw distribution in the material. The macroscopic inelastic deformation is assumed to be due to the nucleation and growth of tensile “wing” micro-cracks associated with frictional sliding on these flaws. Interactions among the cracks are modeled by means of a crack-matrix-effective-medium approach in which each crack experiences a stress field different from that acting on isolated cracks. This yields an effective stress intensity factor at the crack tips which is utilized in the formulation of the crack growth dynamics. Load-induced damage in the material is defined in terms of a scalar crack density parameter, the evolution of which is a function of the existing flaw distribution and the crack growth dynamics. This methodology is applied for the case of uniaxial compression under constant strain rate loading. The model provides a natural prediction of a peak stress (defined as the compressive strength of the material) and also of a transition strain rate, beyond which the compressive strength increases dramatically with the imposed strain rate. The influences of the crack growth dynamics, the initial flaw distribution, and the imposed strain rate on the constitutive response and the damage evolution are studied. It is shown that different characteristics of the flaw distribution are dominant at different imposed strain rates: at low rates the spread of the distribution is critical, while at high strain rates the total flaw density is critical.     

NUMERICAL SIMULATION OF PARTICLE MOTION IN TURBO CLASSIFIER

Research on the flow field inside a turbo classifier is complicated though important. According to the stochastic trajectory model of particles in gas-solid two-phase flow, and adopting the PHOENICS code, numerical simulation is carried out on the flow field, including particle trajectory, in the inner cavity of a turbo classifier, using both straight and backward crooked elbow blades. Computation results show that when the backward crooked elbow blades are used, the mixed stream that passes through the two blades produces a vortex in the positive direction which counteracts the attached vortex in the opposite direction due to the high-speed turbo rotation, making the flow steadier, thus improving both the grade efficiency and precision of the turbo classifier. This research provides positive theoretical evidences for designing sub-micron particle classifiers with high efficiency and accuracy.     

Effect of elastic wall motion on oscillatory flow of micropolar fluid in an annular tube

Summary Oscillatory flow of a micropolar fluid in an annular tube is investigated. The outer wall of the tube is taken to be elastic and the variation in the diameter of the elastic wall due to pulsatile nature of pressure gradient is assumed to be small. The wall motion is governed by a tube law. The nonlinear equations governing the fluid flow and the tube law are solved using perturbation analysis. The steady-streaming phenomenon due to the interaction of convected inertia with viscous effects is studied. The analysis, is carried out for zero mean flow rate. It presents the effects of the elastic nature of the wall combined with micropolar fluid parameters on the mean pressure gradient and wall shear stress for different catheter sizes and frequency parameters. It is found that the effect of micropolarity is of considerable importance for small steady-streaming Reynolds number. Also, it is observed that the relationship between mean pressure gradient and the flow rate depends on the amplitude of the diameter variation, flow rate waveforms and the phase difference between them.     

Energy dynamics in a turbulent channel flow using the Karhunen‐Loéve approach

The dynamical equations for the energy in a turbulent channel flow have been developed by using the Karhunen‐Loéve modes to represent the velocity field. The energy balance equations show that all the energy in the flow originates from the applied pressure gradient acting on the mean flow. Energy redistribution occurs through triad interactions, which is basic to understanding the dynamics. Each triad interaction determines the rate of energy transport between source and sink modes via a catalyst mode. The importance of the proposed method stems from the fact that it can be used to determine both the rate of energy transport between modes as well as the direction of energy flow. The effectiveness of the method in determining the mechanisms by which the turbulence sustains itself is demonstrated by performing a detailed analysis of triad interactions occurring during a turbulent burst in a minimal channel flow. The impact on flow modification is discussed. Copyright © 2002 John Wiley & Sons, Ltd.     

A simple plasticity model for prediction of non-coaxial flow of sand

A bounding surface plasticity model for non-coaxiality, another aspect of anisotropic behavior of sands under rotation of principal stress axes; is developed in the critical state framework. Numerous experimental evidences exist that corroborate dependence of plastic shear strain rate direction on inherent fabric anisotropy. At first, general expressions for plastic strain rate with respect to possible emerge of non-coaxial flow are obtained. Consequently, using an anisotropy state parameter that is specially developed for this model and accounts for the interaction between imposed loading and soil fabric; effect of anisotropy on plastic flow direction is taken into account. Besides, novel circumstances are proposed for plastic modulus and dilatancy under rotation of principal stress axes. Finally, it is shown that the model is able to simulate successfully the non-coaxial behavior of sands subjected to principal stress axes rotation.     

Influence of large eddies on the suspension of solid particles

The phenomena of solid particles suspensions, in a turbulent flow, can more conveniently be described by stochastic models than by diffusion models, particularly in the case of relatively coarse particles.

The fundamental difficulties of using such models are principally due to the difficulty of performing direct measurements of probabilities, because the number of observations (or tests) necessary to obtain physically representative values is important (theoretically infinite).

We have used such a model to describe the movement of spheres in an inclinable pipe.

To do so, we have identified the movement through a Markov process which permits us to show that we can characterize it by the limit distribution for passage probabilities in a cross section. We have used a special system of close-circuit television to measure it, doing a sufficiently large number of observations for the measurements to be significant.

In the case of a vertical pipe, the phenomena is one-dimensional. By using the model stochastic displacement, we obtain a differential equation which it is possible to integrate by assuming an obviously constant radial dispersion. The interpretation of limit distributions for passage probabilities and visual observations of particles movement in the pipe have caused us to conclude that the mean displacment is due, on one hand, to a radial acceleration bounded to a stochastic rotation of the flow and, on the other hand, to the effect of the mean velocity gradient. The experimental results show that the radial dispersion is a function of the relative dimension of particles with respect to the macroscale of the turbulence.

In the case of an inclined pipe, a two-dimensional stochastic model of the displacement is possible, but the integration of the equation is quite complicated and may be done numerically. We have prefered a two-dimensional simulation model. The results of the simulations permit us to obtain a limit repartition of passage probabilities, the moments of which we have compared with those that we have measured. These comparisons show that the model obviously represents the phenomena when the pipe is horizontal or very slightly inclined but differs in the near vertical case. This is due to the simplicity of the model in which we neglect the radial acceleration we have considered previously and the effect of which is negligible in comparison with gravity when the pipe is inclined.

The interpretation of the measurements by comparison of moments with the two-dimensional model shows that the angular dispersion of solid particles is essentially due to big eddies and that the particle diameters are not essential parameters in this case.

By associating this conclusion with that obtained previously concerning the radial dispersion, it seems that the eddies bigger than the macroscale of turbulence may be of capital importance in the dispersion of solid particles and that it will be of practical interest to characterize them as a function of a mean parameter of the flow.

The study of the movement of sufficiently large particles seems to be a method which is able to give this result.     

Reynolds Stress Budgets in Couette and Boundary Layer Flows

Reynolds stress budgets for both Couette and boundary layer flows are evaluated and presented. Data are taken from direct numerical simulations of rotating and non-rotating plane turbulent Couette flow and turbulent boundary layer with and without adverse pressure gradient. Comparison of the total shear stress for the two types of flows suggests that the Couette case may be regarded as the high Reynolds number limit for the boundary layer flow close to the wall. The limit values of turbulence statistics close to the wall for the boundary layer for increasing Reynolds number approach the corresponding Couette flow values. The direction of rotation is chosen so that it has a stabilizing effect, whereas the adverse pressure gradient is destabilizing. The pressure-strain rate tensor in the Couette flow case is presented for a split into slow, rapid and Stokes terms. Most of the influence from rotation is located to the region close to the wall, and both the slow and rapid parts are affected. The anisotropy for the boundary layer decreases for higher Reynolds number, reflecting the larger separation of scales, and becomes close to that for Couette flow. The adverse pressure gradient has a strong weakening effect on the anisotropy. All of the data presented here are available on the web [36]. This revised version was published online in July 2006 with corrections to the Cover Date.     

Rotating electroosmotic flow in a non-uniform microchannel

An analytical model based on lubrication approximation is developed for rotating electroosmotic flow in a narrow slit channel, of which the wall shape and surface potential may vary slowly in the direction of applied fields. The primary and secondary flow fields and the induced pressure gradient, which vary periodically with axial position owing to the gradually varied channel height and surface potentials, are deduced as functions of the inverse Ekman number and the Debye parameter. By studying some limiting cases of special interest, the combined effects of system rotation and the interaction between the geometrical and potential variations are investigated. It is shown that non-uniformity in the channel height and wall potential can qualitatively modify the relationship between system rotation and the primary and secondary flow rates.     

Modelling of rapid pressure-strain in Reynolds stress closures — Difficulties associated with rotational mean flows

Intercomponent energy transfer within the context of Reynolds stress closures is studied. Attention is focussed on the rapid limit of homogeneous flow situations where this energy transfer is caused solely by the rapid pressure strain rate. We present and analyze the performance of the recently proposed rapid pressure strain rate model of Johansson & Hallbäck (J Fluid Mech. 1994) in various homogeneous (rapid) flow situations, and compare with results obtained with other models from the literature and rapid distortion solutions. The prediction difficulties associated with rotational mean flows are analyzed. A generally formulated test case, which as special cases comprises, e.g. plane strain and homogeneous shear flow, is used to illustrate the modelling difficulties associated with rotational mean flows. An axisymmetric case is used to demonstrate that parts of the spectrum with anti-reflectional symmetry, which are instrumental for the dynamics when rotational effects are present, are totally missed in classical Reynolds stress closures. A closer prediction in cases with strong influence of rotation would require introduction of other transported quantities.     

Experimental study of turbulent thermal diffusion in oscillating grids turbulence

We have experimentally detected a new effect, turbulent thermal diffusion, as predicted theoretically by Elperin et al. (Phys Rev Lett (1996) 76:224–228) and associated with the turbulent transport of inertial particles. The essence of this effect is an appearance of a non-diffusive mean flux of particles in the direction of the mean heat flux. This results in formation of large-scale inhomogeneities in the spatial distribution of inertial particles that are accumulated in regions of minimum mean temperature in the surrounding fluid. The experiments were performed in oscillating grids turbulence with an imposed mean temperature gradient. We used Particle Image Velocimetry to determine the turbulent velocity field, and an Image Processing Technique based on the analysis of the intensity of the Mie scattering to determine the spatial distribution of tracer particles. Analysis of the intensity of laser light Mie scattering by tracer particles showed that the tracer particles accumulate in the vicinity of the minimum of the mean temperature. The latter finding confirms the existence of the effect of turbulent thermal diffusion.     

Permeability mapping in porous media by magnetization prepared centric-scan SPRITE

The ability of porous media to transmit fluids is commonly referred to as permeability. The concept of permeability is central for hydrocarbon recovery from petroleum reservoirs and for studies of groundwater flow in aquifers. Spatially resolved measurements of permeability are of great significance for fluid dynamics studies. A convenient concept of local Darcy’s law is suggested for parallel flow systems. The product of porosity and mean velocity images in the plane across the average flow direction is directly proportional to permeability. Single Point Ramped Imaging with T ₁ Enhancement (SPRITE) permits reliable quantification of local fluid content and flow in porous media. It is particularly advantageous for reservoir rocks characterized by fast magnetic relaxation of a saturating fluid. Velocity encoding using the Cotts pulsed field gradient scheme improves the accuracy of measured flow parameters. The method is illustrated through measurements of 2D permeability maps in a capillary bundle, glass bead packs and composite sandstone samples.