Gravity flow of cohesionless granular materials in chutes and channels

1979 ◽  
Vol 92 (1) ◽  
pp. 53-96 ◽  
Author(s):  
Stuart B. Savage
Keyword(s):  
Granular Materials ◽  
Constitutive Relations ◽  
Side Wall ◽  
Wall Friction ◽  
Hydraulic Jumps ◽  
Two Dimensional ◽  
High Deformation ◽  
Gravity Flows ◽  
Theoretical Predictions ◽  
Set Up

A constitutive equation appropriate for flow of cohesionless granular materials at high deformation rates and low stress levels is proposed. It consists of an extension and a reinterpretation of the theory of Goodman & Cowin (1972), and accounts for the non-Newtonian nature of the flow as evidenced by Bagnold's (1954) experiments. The theory is applied to analyses of gravity flows in inclined chutes and vertical channels. Experiments were set up in an attempt to generate two-dimensional shear flows corresponding to these analyses. Velocity profiles measured by a technique which makes use of fibre optic probes agree qualitatively with the theoretical predictions, but direct comparison is inappropriate because of unavoidable side-wall friction effects in the experiments. The existing measure of agreement suggests that the most prominent effects have been included in the proposed constitutive relations. Tests in the inclined chute revealed the possible existence of surge waves and granular jumps analogous to hydraulic jumps.

Maximum Entropy Methods in the Mechanics of Quasi-Static Deformation of Granular Materials

2002 ◽  
Author(s):  
N. P. Kruyt ◽  
L. Rothenburg
Keyword(s):  
Probability Density ◽  
Maximum Entropy ◽  
Granular Materials ◽  
Probability Density Functions ◽  
Contact Forces ◽  
Static Deformation ◽  
Density Functions ◽  
Two Dimensional ◽  
Entropy Methods ◽  
Theoretical Predictions

In statistical physics of dilute gases maximum entropy methods are widely used for theoretical predictions of macroscopic quantities in terms of microscopic quantities. In this study an analogous approach to the mechanics of quasi-static deformation of granular materials is proposed. The reasoning is presented that leads to the definition of an entropy that is appropriate to quasi-static deformation of granular materials. This entropy is formulated in terms of contact quantities, since contacts constitute the relevant microscopic level for granular materials that consist of semirigid particles. The proposed maximum entropy approach is then applied to two cases. The first case deals with the probability density functions of contact forces in a two-dimensional assembly with frictional contacts under prescribed hydrostatic stress. The second case deals with the elastic behaviour of two-dimensional assemblies of non-rotating particles with bonded contacts. For both cases the probability density functions of contact forces are determined from the proposed maximum entropy method, under the constraints appropriate to the case. These constraints form the macroscopic information available about the system. With the probability density functions for contact forces thus determined, theoretical predictions of macroscopic quantities can be made. These theoretical predictions are then compared with results obtained from two-dimensional Discrete Element simulations and from experiments.

Incremental constitutive relations for granular materials based on micromechanics

1993 ◽  
Vol 441 (1913) ◽  
pp. 433-463 ◽  
Keyword(s):  
Granular Materials ◽  
Constitutive Equations ◽  
Constitutive Relations ◽  
Rate Of Change ◽  
Contact Forces ◽  
Two Dimensional ◽  
Microstructural Parameters ◽  
Two Dimensional Flow ◽  
History Of ◽  
Shearing Deformation

Micromechanically based constitutive relations for two-dimensional flow of granular materials are presented. First, overall stresses are related to the interparticle forces and microstructural parameters. Then, the overall velocity gradient is related to measures of relative sliding and rotation of granules. The notion of the class of granules with continuously evolving distribution of contact normals, is introduced. Simple local constitutive relations are considered for the rate of change of the contact forces, the evolution of the contact normals, the mechanism of local failure, and the density of contacts in a particular class. This leads to macroscopic rate constitutive equations through a Taylor averaging method. Due to nonlinearity of the rate constitutive equations, the response is computed by an incremental procedure. As an illustration, the overall response of a two-dimensional assembly of discs subjected to an overall shearing deformation is determined. In addition, explicit results are presented for the evolution of fabric, contact forces, and the history of active and inactive classes of contacts. The stress-strain relations and the evolution of fabric and contact forces are in qualitative agreement with the observed behaviour of granular materials. In light of these results, the mechanisms of failure and inelastic deformation of dense as well as loose granular materials are discussed.

A Method for Testing Cascades with Converging Sidewalls

1978 ◽  
Vol 100 (3) ◽  
pp. 457-464
Author(s):  
G. Meauze´
Keyword(s):  
Experimental Data ◽  
Side Wall ◽  
Local Area ◽  
Computation Method ◽  
Simple Computation ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
Set Up ◽  
Lateral Walls ◽  
Comparison With Experimental Data

Cascade tests on a set-up equipped with converging lateral walls are more representative of the flow in a section of actual turbomachine than in those with two-dimensional-flow. The experiment is carried out at mid-span, where it is necessary to know the local area contraction evolution in order to get more representative data: for this purpose, a simple computation method has been developed and successfully applied, as shown by the comparison with experimental data. By means of successive approaches, the side wall shape can be defined so that a given area contraction evolution can be obtained at the center.

Micromechanical Definition of the Strain Tensor for Granular Materials

1996 ◽  
Vol 63 (3) ◽  
pp. 706-711 ◽  
Author(s):  
N. P. Kruyt ◽  
L. Rothenburg
Keyword(s):  
Granular Materials ◽  
Computer Simulations ◽  
Elastic Moduli ◽  
Contact Stiffness ◽  
Constitutive Relations ◽  
Strain Tensor ◽  
Contact Forces ◽  
Two Dimensional ◽  
Wide Range ◽  
Static Conditions

In order to develop constitutive relations for granular materials from the micromechanical viewpoint, general expressions relating macroscopic stress and strain to contact forces and particle displacements are required. Such an expression for the stress tensor under quasi-static conditions is well established in the literature, but a corresponding expression for the strain tensor has been lacking so far. This paper presents such an expression for two-dimensional assemblies. This expression is verified by computer simulations of biaxial and shear tests. As a demonstration of the use of the developed expression, a study is made of the elastic moduli of two-dimensional, isotropic assemblies of bonded, nonrotating disks. Theoretical expressions are given for the elastic moduli in terms of micromechanical parameters, such as coordination number and contact stiffnesses. Comparison with the results from computer simulations show that the agreement is fairly good over a wide range of coordination numbers and contact stiffness ratios.

Wake Flow of Single and Multiple Yawed Cylinders

2004 ◽  
Vol 126 (5) ◽  
pp. 861-870 ◽  
Author(s):  
A. Thakur ◽  
X. Liu ◽  
J. S. Marshall
Keyword(s):  
Computational Study ◽  
Three Dimensional ◽  
Side Wall ◽  
Wake Flow ◽  
Upstream Region ◽  
Two Dimensional ◽  
Cylinder Wake ◽  
Dimensional Approximation ◽  
The Face ◽  
Drag And Lift

An experimental and computational study is performed of the wake flow behind a single yawed cylinder and a pair of parallel yawed cylinders placed in tandem. The experiments are performed for a yawed cylinder and a pair of yawed cylinders towed in a tank. Laser-induced fluorescence is used for flow visualization and particle-image velocimetry is used for quantitative velocity and vorticity measurement. Computations are performed using a second-order accurate block-structured finite-volume method with periodic boundary conditions along the cylinder axis. Results are applied to assess the applicability of a quasi-two-dimensional approximation, which assumes that the flow field is the same for any slice of the flow over the cylinder cross section. For a single cylinder, it is found that the cylinder wake vortices approach a quasi-two-dimensional state away from the cylinder upstream end for all cases examined (in which the cylinder yaw angle covers the range 0⩽ϕ⩽60°). Within the upstream region, the vortex orientation is found to be influenced by the tank side-wall boundary condition relative to the cylinder. For the case of two parallel yawed cylinders, vortices shed from the upstream cylinder are found to remain nearly quasi-two-dimensional as they are advected back and reach within about a cylinder diameter from the face of the downstream cylinder. As the vortices advect closer to the cylinder, the vortex cores become highly deformed and wrap around the downstream cylinder face. Three-dimensional perturbations of the upstream vortices are amplified as the vortices impact upon the downstream cylinder, such that during the final stages of vortex impact the quasi-two-dimensional nature of the flow breaks down and the vorticity field for the impacting vortices acquire significant three-dimensional perturbations. Quasi-two-dimensional and fully three-dimensional computational results are compared to assess the accuracy of the quasi-two-dimensional approximation in prediction of drag and lift coefficients of the cylinders.

scholarly journals Energy budget in internal wave attractor experiments

2019 ◽  
Vol 880 ◽  
pp. 743-763 ◽  
Author(s):  
Géraldine Davis ◽  
Thierry Dauxois ◽  
Timothée Jamin ◽  
Sylvain Joubaud
Keyword(s):  
Velocity Field ◽  
Internal Wave ◽  
Boundary Layers ◽  
Energy Budget ◽  
Dissipation Rate ◽  
Theoretical Expression ◽  
Two Dimensional ◽  
Energy Sink ◽  
Set Up ◽  
Different Sources

The current paper presents an experimental study of the energy budget of a two-dimensional internal wave attractor in a trapezoidal domain filled with uniformly stratified fluid. The injected energy flux and the dissipation rate are simultaneously measured from a two-dimensional, two-component, experimental velocity field. The pressure perturbation field needed to quantify the injected energy is determined from the linear inviscid theory. The dissipation rate in the bulk of the domain is directly computed from the measurements, while the energy sink occurring in the boundary layers is estimated using the theoretical expression for the velocity field in the boundary layers, derived recently by Beckebanze et al. (J. Fluid Mech., vol. 841, 2018, pp. 614–635). In the linear regime, we show that the energy budget is closed, in the steady state and also in the transient regime, by taking into account the bulk dissipation and, more importantly, the dissipation in the boundary layers, without any adjustable parameters. The dependence of the different sources on the thickness of the experimental set-up is also discussed. In the nonlinear regime, the analysis is extended by estimating the dissipation due to the secondary waves generated by triadic resonant instabilities, showing the importance of the energy transfer from large scales to small scales. The method tested here on internal wave attractors can be generalized straightforwardly to any quasi-two-dimensional stratified flow.

The Reduction of Bias Error in Transfer Function Estimates Using FFT-Based Analyzers

1984 ◽  
Vol 106 (1) ◽  
pp. 29-35 ◽  
Author(s):  
P. Cawley
Keyword(s):  
Transfer Function ◽  
Random Excitation ◽  
Analogue Computer ◽  
Bias Error ◽  
Testing Time ◽  
Theoretical Predictions ◽  
Set Up ◽  
Modal Mass ◽  
Time Required ◽  
Good Agreement

The susceptibility to bias error of two methods for computing transfer (frequency response) functions from spectra produced by FFT-based analyzers using random excitation has been investigated. Results from tests with an FFT analyzer on a single degree-of-freedom system set up on an analogue computer show good agreement with the theoretical predictions. It has been shown that, around resonance, the bias error in the transfer function estimate H2 (Syy/Sxy*) is considerably less than that in the more commonly used estimate, H1 (Sxy/Sxx). The record length, and hence the testing time, required for a given accuracy is reduced by over 50 percent if the H2 calculation procedure is used. The analysis has also shown that if shaker excitation is used on lightly damped structures with low modal mass, it is important to minimize the mass of the force gage and the moving element of the shaker.

Interactions between steady and oscillatory convection in mushy layers

2010 ◽  
Vol 645 ◽  
pp. 411-434 ◽  
Author(s):  
PETER GUBA ◽  
M. GRAE WORSTER
Keyword(s):  
Standing Waves ◽  
Mushy Layer ◽  
Two Dimensional ◽  
Oscillatory Convection ◽  
Mushy Layers ◽  
Theoretical Predictions ◽  
Convection Patterns ◽  
The Stability ◽  
Nonlinear Solutions ◽  
Steady Convection

We study nonlinear, two-dimensional convection in a mushy layer during solidification of a binary mixture. We consider a particular limit in which the onset of oscillatory convection just precedes the onset of steady overturning convection, at a prescribed aspect ratio of convection patterns. This asymptotic limit allows us to determine nonlinear solutions analytically. The results provide a complete description of the stability of and transitions between steady and oscillatory convection as functions of the Rayleigh number and the compositional ratio. Of particular focus are the effects of the basic-state asymmetries and non-uniformity in the permeability of the mushy layer, which give rise to abrupt (hysteretic) transitions in the system. We find that the transition between travelling and standing waves, as well as that between standing waves and steady convection, can be hysteretic. The relevance of our theoretical predictions to recent experiments on directionally solidifying mushy layers is also discussed.

Errata

1993 ◽  
Vol 443 (1919) ◽  
pp. 625-627
Keyword(s):  
Granular Materials ◽  
Constitutive Relations

Proc. R. Soc. Lond . A 441, 433-463 (1993) Incremental constitutive relations for granular materials based on micromechanics By M. M. Mehra Badi, B. Loret and S. Nemat - Nasser Figures 6 and 10 in this paper were originally printed with an incorrect layout. They are printed below, with their correct layout, complete with captions, which remain unchanged.

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