scholarly journals A two-dimensional depth-averaged -rheology for dense granular avalanches

2015 ◽  
Vol 787 ◽  
pp. 367-395 ◽  
Author(s):  
J. L. Baker ◽  
T. Barker ◽  
J. M. N. T. Gray
Keyword(s):  
Velocity Profile ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Theoretical Models ◽  
Quantitative Agreement ◽  
Surface Velocity ◽  
Momentum Balance ◽  
Test Case ◽  
Two Dimensional ◽  
Uniform Solution

Steady uniform granular chute flows are common in industry and provide an important test case for new theoretical models. This paper introduces depth-integrated viscous terms into the momentum-balance equations by extending the recent depth-averaged ${\it\mu}(I)$-rheology for dense granular flows to two spatial dimensions, using the principle of material frame indifference or objectivity. Scaling the cross-slope coordinate on the width of the channel and the velocity on the one-dimensional steady uniform solution, we show that the steady two-dimensional downslope velocity profile is independent of scale. The only controlling parameters are the channel aspect ratio, the slope inclination angle and the frictional properties of the chute and the sidewalls. Solutions are constructed for both no-slip conditions and for a constant Coulomb friction at the walls. For narrow chutes, a pronounced parabolic-like depth-averaged downstream velocity profile develops. However, for very wide channels, the flow is almost uniform with narrow boundary layers close to the sidewalls. Both of these cases are in direct contrast to conventional inviscid avalanche models, which do not develop a cross-slope profile. Steady-state numerical solutions to the full three-dimensional ${\it\mu}(I)$-rheology are computed using the finite element method. It is shown that these solutions are also independent of scale. For sufficiently shallow channels, the depth-averaged velocity profile computed from the full solution is in excellent agreement with the results of the depth-averaged theory. The full downstream velocity can be reconstructed from the depth-averaged theory by assuming a Bagnold-like velocity profile with depth. For wide chutes, this is very close to the results of the full three-dimensional calculation. For experimental validation, a laser profilometer and balance are used to determine the relationship between the total mass flux in the chute and the flow thickness for a range of slope angles and channel widths, and particle image velocimetry (PIV) is used to record the corresponding surface velocity profiles. The measured values are in good quantitative agreement with reconstructed solutions to the new depth-averaged theory.

Stability of a compressible two-dimensional vortex under a three-dimensional perturbation

1984 ◽  
Vol 392 (1803) ◽  
pp. 279-299 ◽  
Keyword(s):  
Mach Number ◽  
Velocity Profile ◽  
Incompressible Flow ◽  
Compressible Fluid ◽  
Three Dimensional ◽  
Critical Value ◽  
Two Dimensional ◽  
Rankine Vortex ◽  
The Core ◽  
Critical Boundary

Kelvin showed that a two-dimensional vortex under a two-dimensional disturbance in incompressible flow responds at a discrete set of eigenvalues, which were found by Broadbent & Moore ( Phil. Trans. R. Soc. Lond. A 290, 353-371 (1979) to become unstable in a compressible fluid. It is now shown that three-dimensional perturbations are also unstable provided the wavelength is greater than some critical value that depends on the Mach number of the vortex. A critical boundary dividing stable from unstable modes is defined. Most of the results relate to a Rankine vortex, as in the previous work mentioned above, but some results are also given for a vortex with a different velocity profile within the core; qualitatively the same kind of behaviour is found.

Fan Similarity Model for the Fan–Intake Interaction Problem

2017 ◽  
Vol 140 (5) ◽  
Author(s):  
Mauro Carnevale ◽  
Feng Wang ◽  
Anthony B. Parry ◽  
Jeffrey S. Green ◽  
Luca di Mare
Keyword(s):  
Numerical Solutions ◽  
Mutual Influence ◽  
Three Dimensional ◽  
Test Case ◽  
Present Contribution ◽  
Shock System ◽  
Aero Engines ◽  
Linearized Model ◽  
Bypass Ratio ◽  
Very High

Very high bypass ratio turbofans with large fan tip diameter are an effective way of improving the propulsive efficiency of civil aero-engines. Such engines, however, require larger and heavier nacelles, which partially offset any gains in specific fuel consumptions. This drawback can be mitigated by adopting thinner walls for the nacelle and by shortening the intake section. This binds the success of very high bypass ratio technologies to the problem of designing an intake with thin lips and short diffuser section, which is well matched to a low speed fan. Consequently, the prediction of the mutual influence between the fan and the intake flow represents a crucial step in the design process. Considerable effort has been devoted in recent years to the study of models for the effects of the fan on the lip stall characteristics and the operability of the whole installation. The study of such models is motivated by the wish to avoid the costs incurred by full, three-dimensional (3D) computational fluid dynamics (CFD) computations. The present contribution documents a fan model for fan–intake computations based on the solution of the double linearization problem for unsteady, transonic flow past a cascade of aerofoils with finite mean load. The computation of the flow in the intake is reduced to a steady problem, whereas the computation of the flow in the fan is reduced to one steady problem and a set of solutions of the linearized model in the frequency domain. The nature of the approximations introduced in the fan representation is such that numerical solutions can be computed inexpensively, while the main feature of the flow in the fan passage, namely the shock system and an approximation of the unsteady flow encountered by the fan are retained. The model is applied to a well-documented test case and compares favorably with much more expensive 3D, time-domain computations.

scholarly journals Nonlinear evolution of interacting oblique waves on two-dimensional shear layers

1989 ◽  
Vol 207 ◽  
pp. 97-120 ◽  
Author(s):  
M. E. Goldstein ◽  
S.-W. Choi
Keyword(s):  
Wave Amplitude ◽  
Numerical Solutions ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Shear Layers ◽  
Instability Wave ◽  
Two Dimensional ◽  
Instability Waves ◽  
Downstream Distance ◽  
Parallel Streams

We consider the effects of critical-layer nonlinearity on spatially growing oblique instability waves on nominally two-dimensional shear layers between parallel streams. The analysis shows that three-dimensional effects cause nonlinearity to occur at much smaller amplitudes than it does in two-dimensional flows. The nonlinear instability wave amplitude is determined by an integro-differential equation with cubic-type nonlinearity. The numerical solutions to this equation are worked out and discussed in some detail. We show that they always end in a singularity at a finite downstream distance.

A Comparative Study Between Two-Dimensional and Three-Dimensional Simulation Results of Melting Inside a Container

2020 ◽  
Author(s):  
Nan Hu ◽  
Li-Wu Fan
Keyword(s):  
Velocity Profile ◽  
Heating Temperature ◽  
Three Dimensional ◽  
Detailed Comparison ◽  
Melting Rate ◽  
Two Dimensional ◽  
End Effects ◽  
Wall Superheat ◽  
Simulation Results ◽  
Dimensional Simulation

Abstract Bother two-dimensional (2D) and three-dimensional (3D) simulations on two example melting problems, i.e., melting in a differentially-heated rectangular cavity and constrained melting in a horizontal cylindrical capsule, were carried out to investigate the rationality of 2D simplification. The effects of thermophysical properties of the phase change material, size of the container along the direction perpendicular to the 2D cross-section, as well as wall superheat were taken into consideration for a systematic and detailed comparison. It was shown that a small length of the container perpendicular to 2D plane will result in a confine space to limit the development of velocity distribution (i.e., parabolic velocity profile) due to the end effects, leading to to an almost identical melting rate to that obtained by the 2D simplified case. A larger size indicates stronger thermal convection (bulk uniform velocity profile) and faster melting rate. When fixing a large size of the container perpendicular to the 2D plane, decreasing the heating temperature and increasing the viscosity of liquid PCM (e.g., by adding nanoparticles) reduce the discrepancy between 2D and 3D simulation results.

Singularities in solutions of the three-dimensional laminar-boundary-layer equations

1985 ◽  
Vol 160 ◽  
pp. 257-279 ◽  
Author(s):  
James C. Williams
Keyword(s):  
Boundary Layer ◽  
Laminar Boundary Layer ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Unsteady Boundary Layer ◽  
Boundary Layer Flows ◽  
Boundary Layer Equations ◽  
New Type ◽  
Steady Laminar

The three-dimensional steady laminar-boundary-layer equations have been cast in the appropriate form for semisimilar solutions, and it is shown that in this form they have the same structure as the semisimilar form of the two-dimensional unsteady laminar-boundary-layer equations. This similarity suggests that there may be a new type of singularity in solutions to the three-dimensional equations: a singularity that is the counterpart of the Stewartson singularity in certain solutions to the unsteady boundary-layer equations.A family of simple three-dimensional laminar boundary-layer flows has been devised and numerical solutions for the development of these flows have been obtained in an effort to discover and investigate the new singularity. The numerical results do indeed indicate the existence of such a singularity. A study of the flow approaching the singularity indicates that the singularity is associated with the domain of influence of the flow for given initial (upstream) conditions as is prescribed by the Raetz influence principle.

Applying Ensemble Kalman Filter to Transonic Flows Through a Two-Dimensional Turbine Cascade

2021 ◽  
Vol 143 (12) ◽  
Author(s):  
Sasuga Ito ◽  
Masato Furukawa ◽  
Kazutoyo Yamada ◽  
Kaito Manabe
Keyword(s):  
Kalman Filter ◽  
Ensemble Kalman Filter ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Optimization Approach ◽  
Two Dimensional ◽  
Turbine Cascade ◽  
Eddy Simulation

Abstract Turbulence is one of the most important phenomena in fluid dynamics. Large eddy simulation (LES) generally allows us to analyze smaller eddies than when using simulations based on unsteady Reynolds-averaged Navier–Stokes equations (URANS). In addition, the numerical solutions of LES show good agreements with experiments and numerical solutions based on direct numerical simulation. URANS simulations are, however, frequently used in academia and industry because LES computations are much more expensive compared with URANS simulations. In this investigation, an optimization of unsolved coefficients of the k–ω two equations model is performed on the transonic flow around T106A low-pressure turbine cascade to improve the accuracy of turbulence prediction with URANS. For the optimization approach, two-dimensional URANS is combined with ensemble Kalman filter which is one of the data assimilation techniques. In the assimilation process, a time- and spanwise-averaged LES result is used as pseudo-experimental data. Three-dimensional URANS simulations are performed for the evaluation of the optimization effect. URANS simulations are also applied to a different turbine cascade flow for the evaluation of the robustness of the optimized coefficients. These URANS results confirmed that the optimized coefficients improve the accuracy of turbulence prediction.

scholarly journals Forming three-dimensional closed shapes from two-dimensional soft ribbons by controlled buckling

2018 ◽  
Vol 5 (2) ◽  
pp. 171962 ◽  
Author(s):  
Michio Aoki ◽  
Jia-Yang Juang
Keyword(s):  
Three Dimensional ◽  
Quantitative Agreement ◽  
Two Dimensional ◽  
Element Analysis ◽  
Hollow Structures ◽  
Strategic Design ◽  
Post Buckling ◽  
Manufacturing Techniques ◽  
New Applications ◽  
3D Shapes

Conventional manufacturing techniques—moulding, machining and casting—exist to produce three-dimensional (3D) shapes. However, these industrial processes are typically geared for mass production and are not directly applicable to residential settings, where inexpensive and versatile tools are desirable. Moreover, those techniques are, in general, not adequate to process soft elastic materials. Here, we introduce a new concept of forming 3D closed hollow shapes from two-dimensional (2D) elastic ribbons by controlled buckling. We numerically and experimentally characterize how the profile and thickness of the ribbon determine its buckled shape. We find a 2D master profile with which various elliptical 3D shapes can be formed. More complex natural and artificial hollow shapes, such as strawberry, hourglass and wheel, can also be achieved via strategic design and pattern engraving on the ribbons. The nonlinear response of the post-buckling regime is rationalized through finite-element analysis, which shows good quantitative agreement with experiments. This robust fabrication should complement conventional techniques and provide a rich arena for future studies on the mechanics and new applications of elastic hollow structures.

A Lumped-Parameter Model for Stagnant Thermal Conductivity of Spatially Periodic Porous Media

1995 ◽  
Vol 117 (2) ◽  
pp. 264-269 ◽  
Author(s):  
C. T. Hsu ◽  
P. Cheng ◽  
K. W. Wong
Keyword(s):  
Thermal Conductivity ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Lumped Parameter Model ◽  
Circular Cylinders ◽  
Parameter Method ◽  
Two Dimensional ◽  
Algebraic Expressions ◽  
Lumped Parameter ◽  
Spatially Periodic

Based on a lumped-parameter method, algebraic expressions for the stagnant thermal conductivity of some two-dimensional and three-dimensional spatially periodic media are obtained. The geometries under consideration include arrays of touching and non-touching in-line square and circular cylinders (two-dimensional), as well as touching and nontouching in-line cubes (three-dimensional). A comparison of results based on these algebraic expressions with existing numerical solutions and experimental data shows that they are in excellent agreement.

On the nature of unsteady three-dimensional laminar boundary-layer separation

1978 ◽  
Vol 88 (2) ◽  
pp. 241-258 ◽  
Author(s):  
James C. Williams
Keyword(s):  
Boundary Layer ◽  
Laminar Boundary Layer ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Boundary Layer Separation ◽  
The Body ◽  
Two Dimensional ◽  
Boundary Layer Equations ◽  
Unsteady Separation ◽  
Similar Solutions

Solutions have been obtained for a family of unsteady three-dimensional boundary-layer flows which approach separation as a result of the imposed pressure gradient. These solutions have been obtained in a co-ordinate system which is moving with a constant velocity relative to the body-fixed co-ordinate system. The flows studied are those which are steady in the moving co-ordinate system. The boundary-layer solutions have been obtained in the moving co-ordinate system using the technique of semi-similar solutions. The behaviour of the solutions as separation is approached has been used to infer the physical characteristics of unsteady three-dimensional separation.In the numerical solutions of the three-dimensional unsteady laminar boundary-layer equations, subject to an imposed pressure distribution, the approach to separation is characterized by a rapid increase in the number of iterations required to obtain converged solutions at each station and a corresponding rapid increase in the component of velocity normal to the body surface. The solutions obtained indicate that separation is best observed in a co-ordinate system moving with separation where streamlines turn to form an envelope which is the separation line, as in steady three-dimensional flow, and that this process occurs within the boundary layer (away from the wall) as in the unsteady two-dimensional case. This description of three-dimensional unsteady separation is a generalization of the two-dimensional (Moore-Rott-Sears) model for unsteady separation.

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