Towards a constitutive law for the unsteady contact stress in granular media

In order to build a unified modelling for granular media by means of Eulerian averaged equations, it is necessary to study two contributions in the effective Cauchy stress tensor: the first one concerns solid and fluid matter, including contact and collisions between grains; the second one focuses on the random movements of grains and fluid, similar to Reynolds stress for turbulent flows. It is shown that the point of view of piecewise continuous media already used for two phase flows allows one to derive a constitutive equation for the first contribution, under the form of a partial differential equation. Similarly as for the Reynolds stress in turbulent flows, this equation cannot be written only in terms of averaged quantities without adequate approximations. The structure of the closed equation is discussed with respect to irreversible thermodynamics, and in connection with some already existing models. It is emphasised that numerical simulations by the discrete elements method can be used in order to validate these approximations, through numerical experiments statistically considered. Finally an extension of this approach to the second contribution of the effective Cauchy stress tensor is briefly discussed, showing how the modelling of both contributions have to be coupled.      

The effect of slip in the flow of a branched PP melt: experiments and simulations

In this paper visualisation and direct velocity profile measurement experiments for a branched polypropylene melt in a 10:1 axisymmetric contraction demonstrate the onset of wall slip. Video processing of the flow shows the formation of vortices and their diminution with increasing flow rate. Numerical simulations using a multimode K-BKZ viscoelastic and a purely viscous (Cross) model—both of them incorporating a nonlinear slip law—were used to predict the flow kinematics and dynamics as well as to deduce the slip velocity function by performing fitting to the velocity profiles. It was found that the numerical predictions agree well with the experimental results for the velocity profiles, and vortex formation, growth and reduction. It is suggested that such experiments (visualisation of entrance flow and direct velocity profile measurement) can be useful in evaluating the validity of constitutive equations and slip laws in the flow of polymer melts through processing equipment.     

Cattaneo-Christov heat flux model for third-grade fluid flow towards exponentially stretching sheet

The Cattaneo-Christov heat flux in the two-dimensional (2D) flow of a third-grade fluid towards an exponentially stretching sheet is investigated. The energy equation is considered through thermal relaxation. Similarity transformations are accounted to obtain the ordinary differential systems. The converted non-dimensional equations are solved for the series solutions. The convergence analysis of the computed solutions is reported. The graphical results of the velocity and temperature profiles are plotted and elaborated in detail. The results show that the thermal relaxation enhances the temperature gradient while reduces the temperature profile.     

Large eddy simulations on the effect of the irregular roughness shape on turbulent channel flows

Large Eddy Simulations (LES) are carried out to investigate on the mean flow in turbulent channel flows over irregular rough surfaces. Here the attention is focused to selectively investigate on the effect induced by crests or cavities of the roughness. The irregular shape is generated through the super-imposition of sinusoidal functions having random amplitude and four different wave-lengths. The irregular roughness profile is reproduced along the spanwise direction in order to obtain a 2D rough shape. The analysis of the mean velocity profiles shows that roughness crests induce higher effect in the outer-region whereas roughness cavities cause the highest effects in the inner-region with a reduced effect in the external region. The numerical simulations have been carried out at friction Reynolds number Re_τ=395. Similar results have been found for the higher order statistics: turbulence intensities or shear stresses. The analysis of the Reynolds stress anisotropy tensor confirms the existence of specific roles of cavities and crests in the turbulence modulation.     

On the consequences of material frame-indifference in algebraic stress models

The principle of material frame-indifference (MFI) is a fundamental and controversial principle of continuum mechanics that has been invoked recently to derive nonlinear algebraic models for stresses of viscoelastic liquids. The purpose of the present study is to identify regions of a flow field where MFI should be considered. Such regions are identified by computing the angular velocity of the principal directions of the rate-of-deformation tensor in order to obtain an Euclidean objective vorticity tensor. An analysis is carried out for uniform shear and extensional flows, and for a Couette flow. The method is then applied to the planar flow through an abrupt 4:1 contraction and to the two-dimensional stream past a circular cylinder. The main results are: (1) MFI should be taken into account in regions characterized by the transition between two different kinematics and a significant velocity magnitude, and (2) MFI can be safely ignored in regions of pure viscometric behaviour as well as in recirculation regions. The consequences of MFI being taken into account are then examined upon using the Euclidean objective vorticity tensor in a simple algebraic constitutive law for viscoelastic fluids.     

A Reynolds stress model for turbulent flow of homogeneous polymer solutions

Using a priori analyses of direct numerical simulation (DNS) data, a Reynolds stress model (RSM) is developed to account for the influence of polymer additives on turbulent flow over a wide range of flow conditions. The Finitely Extensible Nonlinear Elastic-Peterlin (FENE-P) rheological constitutive model is utilized to evaluate the polymer contribution to the stress tensor. Thirteen DNS data sets are used to analyze the budgets of elastic stress–velocity gradient correlations as well as Reynolds stress and dissipation transport. Closures are developed in the framework of the RSM model for all the required unknown and non-linear terms. The polymer stresses, velocity profiles, turbulent flow statistics and the percentage of friction drag reduction predicted by the RSM model are in good agreement with present and those obtained from independent DNS data over a wide range of rheological and flow parameters.     

Reduced rank static error covariance for high‐dimensional applications

A method for creating static (e.g., stationary) error covariance of reduced rank for potential use in hybrid variational‐ensemble data assimilation is presented. The choice of reduced rank versus full rank static error covariance is made in order to allow the use of an improved Hessian preconditioning in high‐dimensional applications. In particular, this method relies on using block circulant matrices to create a high‐dimensional global covariance matrix from a low‐dimensional local sub‐matrix. Although any covariance used in variational data assimilation would be an acceptable choice for the pre‐defined full‐rank static error covariance, for convenience and simplicity, we use a symmetric Topelitz matrix as a prototype of static error covariance. The methodology creates a square root covariance, which has a practical advantage for Hessian preconditioning in reduced rank, ensemble‐based data assimilation. The experiments conducted examine multivariate covariance that includes the impact of cross‐variable correlations, in order to have a more realistic assessment of the value of the constructed static error covariance approximation. The results show that it may be possible to reduce the rank of matrix to O(10) and still obtain an acceptable approximation of the full‐rank static covariance matrix. Copyright © 2016 John Wiley & Sons, Ltd.     

Ultrasonic attenuation in polycrystals using a self-consistent approach

The self-consistent method of averaging elastic moduli to define the effective medium of a polycrystal is used to investigate the dynamic problem of wave propagation. An alternative covariance tensor describing the elastic moduli fluctuations of the polycrystal containing self-consistent elastic properties is derived and found to be significantly smaller than the covariance tensor formed through traditional Voigt averaging. Attenuation curves are generated using the self-consistent elastic moduli and covariance tensors and these results are compared with previous Voigt-averaged estimates. The second-order polycrystalline dispersion relation for the self-consistent scheme is found for cases of low and high crystallite anisotropy. The attenuation coefficients and dispersion relations derived through the self-consistent scheme are considerably different than previous estimates. Experimentally measured longitudinal attenuation coefficients support the use of the self-consistent scheme for estimation of attenuation.     

Flow near the permeable boundary of a porous medium: An experimental investigation using LDA

 Fluid flow at the interface of a porous medium and an open channel is the governing phenomenon in a number of processes of industrial importance. Traditionally, this has been modeled by applying the Brinkman’s modification of Darcy’s law to obtain the velocity profile in terms of an additional parameter known as the “apparent viscosity” or the “slip coefficient”. To test this ad hoc approach, a detailed experimental investigation of the flow was conducted using Laser Doppler Anemometry (LDA) in the close vicinity of the permeable boundary of a porous medium. The porous medium used in the experiments consisted of a network of continuous glass strands woven together in a random fashion. A Hele–Shaw cell was partially filled with a fibrous preform such that an open channel flow is coupled with the Darcy flow inside the preform through the permeable interface of the preform. The open channel portion of the Hele–Shaw cell also acts as an ideal porous medium of known in-plane permeability which is much higher than the permeability of the fibrous porous medium. A viscous fluid is injected at a constant flow rate through the above arrangement and a saturated and steady flow is established through the cell. Using LDA, steady state velocity profiles are accurately measured by traversing across the cell in the direction perpendicular to the flow. A series of experiments were conducted in which fluid viscosity, flow rate, solid volume fraction of the porous medium and depth of the Hele–Shaw cell were varied. For each and every case in which the conditions for Hele–Shaw approximation were valid, the depth of the boundary layer zone or the screening length inside the fibrous preform was found to be of the order of the channel depth. This is much larger as compared to the Brinkman’s prediction of the screening length which is of the order of √K, where K is the permeability of the fibrous porous medium. Based on this finding, we modified the boundary condition in the Brinkman’s solution and found that the velocity profile results compared well with the experimental data for the planar geometry and the fibrous preforms for volume fractions of 7%, 14% and 21% for Hele–Shaw cell depths of 1.6 and 3.175 mm. For a cell depth of 4.8 cm, in which the Hele–Shaw approximation was not valid, the boundary layer thickness or the screening length was found to be less than the mold or channel depth but was still much larger than the Brinkman’s prediction. Received: 10 May 1996 / Accepted: 26 August 1996     

Laser stripe measurements of near-wall solid fraction in channel flows of liquid-granular mixtures

A simple, robust, and accurate imaging technique is proposed to measure granular concentration profiles in channel flows of liquid-granular mixtures. We focus on moderate to high granular concentrations (5–50%), for which optical access is restricted to regions close to a transparent wall. To measure concentrations in this range, we illuminate solid grains moving near the wall using a transverse laser light sheet. The evolving shape of the laser stripe, deformed by passing grains, is then monitored using an oblique camera. Statistics of the granular distance to wall can thus be acquired and converted to volumetric solid fraction measurements. The method is verified using fluidization cell tests and applied to open-channel sheet flow experiments. Free of any parameter adjustment, the laser stripe method is found to yield good results, and allows joint measurements of granular velocity and solid fraction profiles.     

Effective thermal conductivity of transversely isotropic media with arbitrary oriented ellipsoïdal inhomogeneities

The objective of this work is to investigate the thermal conduction phenomena in transversely isotropic geomaterials or rock-like composites with arbitrary oriented ellipsoïdal inhomogeneities of low aspect ratio. Based on the evaluation of the Green function, we provide here new expressions for the interaction tensor whose knowledge permits to obtain the concentration tensor of the polarization field used itself to evaluate the effective thermal conductivity tensor by homogenization. Some particular cases of the obtained general solution are equally presented, in order to validate the developed formalism. The obtained results are next used to study the effect of matrix anisotropy, pores systems and microstructure-related parameters on the overall effective thermal conductivity in transversely isotropic rocks. A two-step homogenization scheme is developed for the prediction of the initial anisotropy effects and to test the ability of the proposed model in the evaluation of effective thermal conductivity. With the help of an Orientation Distribution Function (ODF) the anisotropy due to the pore systems is also accounted. Numerical applications and comparisons with available experimental data are finally carried out for a partially saturated Opalinus clay and an argillite which are both composed of an argillaceous matrix and multiple solid minerals constituents.     

Fiber suspension flow in a tapered channel: The effect of flow/fiber coupling

A numerical model for predicting the flow and orientation state of semi-dilute, rigid fiber suspensions in a tapered channel is presented. The effect of the two-way flow/fiber coupling is investigated for low Reynolds number flow using the constitutive model of Shaqfeh and Fredrickson. An orientation distribution function is used to describe the local orientation state of the suspension and evolves according to a Fokker–Plank type equation. The planar orientation distribution function is determined along streamlines of the flow and is coupled with the fluid momentum equations through a fourth-order orientation tensor. The coupling term accounts for the two-way interaction and momentum exchange between the fluid and fiber phases. The fibers are free to interact through long range hydrodynamic fiber–fiber interactions which are modeled using a rotary diffusion coefficient, an approach outlined by Folgar and Tucker. Numerical predictions are made for two different orientation states at the inlet to the contraction, namely a fully random and a partially aligned fiber orientation state. Results from these numerical predictions show that the streamlines of the flow are altered and that velocity profiles change from Jeffery–Hamel, to something resembling a plug flow when the fiber phase is considered in the fluid momentum equations. This phenomenon was found when the suspension enters the channel in either a pre-aligned, or in a fully random orientation state. When the suspension enters the channel in an aligned orientation state, fiber orientation is shown to be only marginally changed when the two-way coupling is included. However, significant differences between coupled and uncoupled predictions of fiber orientation were found when the suspension enters the channel in a random orientation state. In this case, the suspension was shown to align much more quickly when the mutual coupling was accounted for and profiles of the orientation anisotropy were considerably different both qualitatively and quantitatively.     

Volumetric three-component velocimetry measurements of the turbulent flow around a Rushton turbine

Volumetric three-component velocimetry measurements have been taken of the flow field near a Rushton turbine in a stirred tank reactor. This particular flow field is highly unsteady and three-dimensional, and is characterized by a strong radial jet, large tank-scale ring vortices, and small-scale blade tip vortices. The experimental technique uses a single camera head with three apertures to obtain approximately 15,000 three-dimensional vectors in a cubic volume. These velocity data offer the most comprehensive view to date of this flow field, especially since they are acquired at three Reynolds numbers (15,000, 107,000, and 137,000). Mean velocity fields and turbulent kinetic energy quantities are calculated. The volumetric nature of the data enables tip vortex identification, vortex trajectory analysis, and calculation of vortex strength. Three identification methods for the vortices are compared based on: the calculation of circumferential vorticity; the calculation of local pressure minima via an eigenvalue approach; and the calculation of swirling strength again via an eigenvalue approach. The use of two-dimensional data and three-dimensional data is compared for vortex identification; a ‘swirl strength’ criterion is less sensitive to completeness of the velocity gradient tensor and overall provides clearer identification of the tip vortices. The principal components of the strain rate tensor are also calculated for one Reynolds number case as these measures of stretching and compression have recently been associated with tip vortex characterization. Vortex trajectories and strength compare favorably with those in the literature. No clear dependence of trajectory on Reynolds number is deduced. The visualization of tip vortices up to 140° past blade passage in the highest Reynolds number case is notable and has not previously been shown.     

Turbulent flow in a channel at a low Reynolds number

A two-component laser Doppler velocimeter with high spatial and temporal resolution was used to obtain measurements for fully developed turbulent flow of water through a channel with an aspect ratio of 12 : 1 at Re=5700 (based on the centerline velocity and the half-height of the channel). Statistical quantities that were determined are the mean streamwise velocity, the root-mean-square of the fluctuations of the streamwise and the normal velocities, the Reynolds shear stress and higher order moments. Turbulence production is calculated from these quantities. Turbulence statistics obtained from experiments are compared with results from a direct numerical simulation at the same Reynolds number. The good agreement validates a recent DNS, at Re=5700, which is approximately twice as large as used in most previous studies. Received: 12 May 1997 / Accepted: 8 April 1998     

Removal of unwanted fluid

This work is concerned with the removal of unwanted fluid through the source–sink pair. The source consists of fluid issuing out of a nozzle in the form of a jet and the sink is a pipe that is kept some distance from the source pipe. Of concern is the percentage of source fluid sucked through the sink. The experiments have been carried in a large glass water tank. The source nozzle diameter is 6 mm and the sink pipe diameter is either 10 or 20 mm. The horizontal and vertical separations and angles between these source and sink pipes are adjustable. The flow was visualized using KMnO₄ dye, planer laser induced fluorescence and particle streak photographs. To obtain the effectiveness (that is percentage of source fluid entering the sink pipe), titration method is used. The velocity profiles with and without the sink were obtained using particle image velocimetry. The sink flow rate to obtain a certain effectiveness increase dramatically with lateral separation. The sink diameter and the angle between source and the sink axes don’t influence effectiveness as much as the lateral separation.     

Accuracy of tomographic particle image velocimetry data on a turbulent round jet

Tomographic particle image velocimetry (Tomo-PIV) was applied on a turbulent round air jet to quantitatively assess the accuracy of velocity gradients obtained in the self-similar turbulent region. The jet Reynolds number based on the nozzle diameter (d) was Re_d = 3000. Mean velocity, turbulent intensities, and Reynolds shear stress at the center plane of the jet were measured. In addition, statistical results of Tomo-PIV along the axial direction were assessed by performing a separate set of two-dimensional two-component PIV experiments on a “side view” plane along the jet axis. Moreover, the probability distribution functions of four components of the measured velocity gradients in the axial and radial directions were validated by these “side view” planar PIV data. The root mean square of the velocity divergence values relative to the norm of the velocity gradient tensor was 0.36. Furthermore, the on- and off-diagonal components of the velocity gradients satisfied the axisymmetric isotropy conditions. The divergence error in the data affected only areas with low gradient magnitude. Therefore, turbulent structures in the regions with intense vorticity and dissipation can be closely monitored. On this basis, the joint pdfs of the invariants of the velocity gradient and strain and rotation tensor rates were produced and compared well with those in isotropic turbulence studies.     

Detailed experimental study of a highly compressible supersonic turbulent plane mixing layer and comparison with most recent DNS results: “Towards an accurate description of compressibility effects in supersonic free shear flows”

A compressible supersonic mixing layer at convective Mach number (Mc) equal to 1 has been studied experimentally in a dual stream supersonic/subsonic wind-tunnel. Laser Doppler Velocimetry (L.D.V.) measurements were performed making possible a full estimation of the mean and turbulent 3D velocity fields in the mixing layer. The Reynolds stress tensor was described. In particular, some anisotropy coefficients were obtained. It appears that the structure of the Reynolds tensor is almost not affected by compressibility at least up to Mc = 1.The turbulent kinetic energy budget was also experimentally estimated. Reynolds analogies assumptions were used to obtain density/velocity correlations in order to build the turbulent kinetic energy budget from LDV measurements. Results have been compared to other experimental and numerical results. Compressibility effects on the turbulent kinetic energy budget have been detected and commented. A study about thermodynamics flow properties was also performed using most recent DNS results experimentally validated by the present data. A non-dimensional number is then introduced in order to quantify the real effect of pressure fluctuations on the thermodynamics quantities fluctuations.     

Dispersion of Gas Bubbles in Large-Scale Homogeneous Isotropic Turbulence

An approximate analysis is given of the dispersion of gas bubbles that rise at large Reynolds number through large-scale homogeneous, isotropic turbulence, characterized by the Kraichnan energy-spectrum function. A fairly well-established equation of motion of the bubbles, originally proposed by Thomas et al. [16], is used to derive a closed set of equations for the components of the dispersion tensor of the bubbles in a manner analogous to that used by Saffman [12] for fluid particles and by Pismen and Nir [10, 11] for solid particles. The equations are then solved to obtain the diffusivities and the intensities of bubble velocity fluctuations. Analytical solutions are compared with results from simulations of the bubble motion in a Gaussian random velocity field.     

Survey of multi-wire probe data processing techniques and efficient processing of four-wire probe velocity measurements in turbulent flows

 A general presentation of the two main approaches for the extraction of velocity information from multi-wire hot-wire probe measurements is first given, along with a short introduction to the different data processing techniques currently available. This global survey is intended to help researchers interested in using multi-wire probes (for velocity measurements) choose more easily a data processing technique adapted to their experiments and laboratory facilities. In the second part, a numerically efficient four-wire probe data processing technique, especially suited for flows with relatively high turbulence intensities (up to 30%), is described. With this technique, an accurate solution can be found in almost all of the probe's acceptance domain and the directional sensitivity calibration does not have to be performed in situ. The accuracy of the method is demonstrated with a 3D airfoil near-wake experiment for which four-wire probe measurements of mean velocity, full Reynolds stress tensor and triple correlations are compared with three-component LDA measurements. Received: 25 February 1999/Accepted: 17 September 1999