Rapid granular flows down inclined planar chutes. Part 2. Linear stability analysis of steady flow solutions

2010 ◽  
Vol 652 ◽  
pp. 461-488 ◽  
Author(s):  
MARK J. WOODHOUSE ◽  
ANDREW J. HOGG
Keyword(s):  
Linear Stability ◽  
Granular Flow ◽  
Continuum Model ◽  
Three Dimensional ◽  
Linear Instability ◽  
Steady Flows ◽  
Density Inversion ◽  
Instability Mechanism ◽  
Flow Curves ◽  
Steady Solutions

The linear stability of steady solutions for a rapid granular flow down an inclined chute, modelled using a kinetic theory continuum model, is analysed. The previous studies of Forterre & Pouliquen (J. Fluid Mech., vol. 467, 2002, p. 361) and Mitarai & Nakanishi (J. Fluid Mech., vol. 507, 2004, p. 309) are extended by considering fully three-dimensional perturbations, allowing variations in both the cross-slope and downslope directions, as well as normal to the base. Our results demonstrate the existence of three qualitatively different unstable perturbations, each of which can be the most rapidly growing instability for different steady flows. By considering the linear stability of many steady solutions along macroscopic flow curves, we show that linear stability occurs in only a small part of parameter space, and furthermore the regions of linear instability do not correlate with density inversion of the underlying steady solutions. Our results suggest that inelastic clustering is the dominant instability mechanism.

scholarly journals Parabolized stability analysis of jets from serrated nozzles

2016 ◽  
Vol 789 ◽  
pp. 36-63 ◽  
Author(s):  
Aniruddha Sinha ◽  
Kristján Gudmundsson ◽  
Hao Xia ◽  
Tim Colonius
Keyword(s):  
Linear Stability ◽  
Near Field ◽  
Pressure Fluctuations ◽  
Three Dimensional ◽  
Hydrodynamic Pressure ◽  
Linear Instability ◽  
Base Flow ◽  
Linear Stability Theory ◽  
Instability Wave ◽  
Mean Flow

We study the viscous spatial linear stability characteristics of the time-averaged flow in turbulent subsonic jets issuing from serrated (chevroned) nozzles, and compare them to analogous round jet results. Linear parabolized stability equations (PSE) are used in the calculations to account for the non-parallel base flow. By exploiting the symmetries of the mean flow due to the regular arrangement of serrations, we obtain a series of coupled two-dimensional PSE problems from the original three-dimensional problem. This reduces the solution cost and manifests the symmetries of the stability modes. In the parallel-flow linear stability theory (LST) calculations that are performed near the nozzle to initiate the PSE, we find that the serrated nozzle reduces the growth rates of the most unstable eigenmodes of the jet, but their phase speeds are approximately similar. We obtain encouraging validation of our linear PSE instability wave results vis-à-vis near-field hydrodynamic pressure data acquired on a phased microphone array in experiments, after filtering the latter with proper orthogonal decomposition (POD) to extract the energetically dominant coherent part. Additionally, a large-eddy simulation database of the same serrated jet is investigated, and its POD-filtered pressure field is found to compare favourably with the corresponding PSE solution within the jet plume. We conclude that the coherent hydrodynamic pressure fluctuations of jets from both round and serrated nozzles are reasonably consistent with the linear instability modes of the turbulent mean flow.

scholarly journals On the stability of the μ(I) rheology for granular flow

2017 ◽  
Vol 833 ◽  
pp. 302-331 ◽  
Author(s):  
J. D. Goddard ◽  
Jaesung Lee
Keyword(s):  
Linear Stability ◽  
Granular Flow ◽  
Convective Instability ◽  
Continuum Model ◽  
Shear Bands ◽  
Closed Form Solution ◽  
Static Field ◽  
Form Solution ◽  
Base Flow ◽  
Field Equations

This article deals with the Hadamard instability of the so-called$\unicode[STIX]{x1D707}(I)$model of dense rapidly sheared granular flow, as reported recently by Barkeret al.(J. Fluid Mech., vol. 779, 2015, pp. 794–818). The present paper presents a more comprehensive study of the linear stability of planar simple shearing and pure shearing flows, with account taken of convective Kelvin wavevector stretching by the base flow. We provide a closed-form solution for the linear-stability problem and show that wavevector stretching leads to asymptotic stabilization of the non-convective instability found by Barkeret al.(J. Fluid Mech., vol. 779, 2015, pp. 794–818). We also explore the stabilizing effects of higher velocity gradients achieved by an enhanced-continuum model based on a dissipative analogue of the van der Waals–Cahn–Hilliard equation of equilibrium thermodynamics. This model involves a dissipative hyperstress, as the analogue of a special Korteweg stress, with surface viscosity representing the counterpart of elastic surface tension. Based on the enhanced-continuum model, we also present a model of steady shear bands and their nonlinear stability against parallel shearing. Finally, we propose a theoretical connection between the non-convective instability of Barkeret al.(J. Fluid Mech., vol. 779, 2015, pp. 794–818) and the loss of generalized ellipticity in the quasi-static field equations. Apart from the theoretical interest, the present work may suggest stratagems for the numerical simulation of continuum field equations involving the$\unicode[STIX]{x1D707}(I)$rheology and variants thereof.

Instability of isolated three-dimensional magnetic null points

1998 ◽  
Vol 59 (3) ◽  
pp. 537-541 ◽  
Author(s):  
MANUEL NÚÑEZ
Keyword(s):  
Linear Stability ◽  
Three Dimensional ◽  
Null Point ◽  
Static Equilibrium ◽  
Flux Surface ◽  
Form Part ◽  
Magnetic Null ◽  
Transverse Oscillations

Although most magnetic neutral points occurring in nature seem to form part of a continuum, recent studies of reconnection have centred on static equilibria in the neighbourhood of an isolated three-dimensional null point. The linear stability of this configuration is studied here. It is found that one may choose a flux surface so that transverse oscillations localized around the surface and polarized within it must grow exponentially in time. This means that any static equilibrium containing an isolated three-dimensional null point is linearly unstable.

Secondary motion in convection layers generated by lateral heating of a solute gradient

2002 ◽  
Vol 455 ◽  
pp. 1-19 ◽  
Author(s):  
CHO LIK CHAN ◽  
WEN-YAU CHEN ◽  
C. F. CHEN
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Convection Cell ◽  
Experimental Conditions ◽  
Longitudinal Plane ◽  
Secondary Motion ◽  
Solute Gradient

The three-dimensional motion observed by Chen & Chen (1997) in the convection cells generated by sideways heating of a solute gradient is further examined by experiments and linear stability analysis. In the experiments, we obtained visualizations and PIV measurements of the velocity of the fluid motion in the longitudinal plane perpendicular to the imposed temperature gradient. The flow consists of a horizontal row of counter-rotating vortices within each convection cell. The magnitude of this secondary motion is approximately one-half that of the primary convection cell. Results of a linear stability analysis of a parallel double-diffusive flow model of the actual ow show that the instability is in the salt-finger mode under the experimental conditions. The perturbation streamlines in the longitudinal plane at onset consist of a horizontal row of counter-rotating vortices similar to those observed in the experiments.

STEADY FLOWS OF VISCOUS COMPRESSIBLE FLUIDS IN EXTERIOR DOMAINS UNDER SMALL PERTURBATIONS OF GREAT POTENTIAL FORCES

1993 ◽  
Vol 03 (06) ◽  
pp. 725-757 ◽  
Author(s):  
ANTONÍN NOVOTNÝ
Keyword(s):  
Existence And Uniqueness ◽  
Compressible Flows ◽  
Three Dimensional ◽  
Compressible Fluids ◽  
Steady Flows ◽  
Exterior Domains ◽  
Uniqueness Of Solutions ◽  
Small Perturbations ◽  
Zero Velocity ◽  
Potential Forces

We investigate the steady compressible flows in three-dimensional exterior domains, in R3 and [Formula: see text], under the action of small perturbations of large potential forces and zero velocity at infinity. We prove existence and uniqueness of solutions in L2-spaces, and study their regularity as well as the decay at infinity.

scholarly journals NeXtYZ: Three-dimensional electron density inversion for dynasonde ionograms

Radio Science ◽  
2006 ◽  
Vol 41 (6) ◽  
pp. n/a-n/a ◽  
Author(s):  
N. A. Zabotin ◽  
J. W. Wright ◽  
G. A. Zhbankov
Keyword(s):  
Electron Density ◽  
Three Dimensional ◽  
Density Inversion ◽  
Dimensional Electron
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