scholarly journals Multiple solutions for granular flow over a smooth two-dimensional bump

2017 ◽  
Vol 815 ◽  
pp. 77-116 ◽  
Author(s):  
S. Viroulet ◽  
J. L. Baker ◽  
A. N. Edwards ◽  
C. G. Johnson ◽  
C. Gjaltema ◽  
...  
Keyword(s):  
Initial Conditions ◽  
Pyroclastic Flows ◽  
Curvilinear Coordinates ◽  
Standard Theory ◽  
Small Scale ◽  
Two Dimensional ◽  
Cartesian Coordinates ◽  
Terrain Following ◽  
Coordinate Mapping ◽  
Cartesian Coordinate

Geophysical granular flows, such as avalanches, debris flows, lahars and pyroclastic flows, are always strongly influenced by the basal topography that they flow over. In particular, localised bumps or obstacles can generate rapid changes in the flow thickness and velocity, or shock waves, which dissipate significant amounts of energy. Understanding how a granular material is affected by the underlying topography is therefore crucial for hazard mitigation purposes, for example to improve the design of deflecting or catching dams for snow avalanches. Moreover, the interactions with solid boundaries can also have important applications in industrial processes. In this paper, small-scale experiments are performed to investigate the flow of a granular avalanche over a two-dimensional smooth symmetrical bump. The experiments show that, depending on the initial conditions, two different steady-state regimes can be observed: either the formation of a detached jet downstream of the bump, or a shock upstream of it. The transition between the two cases can be controlled by adding varying amounts of erodible particles in front of the obstacle. A depth-averaged terrain-following avalanche theory that is formulated in curvilinear coordinates is used to model the system. The results show good agreement with the experiments for both regimes. For the case of a shock, time-dependent numerical simulations of the full system show the evolution to the equilibrium state, as well as the deposition of particles upstream of the bump when the inflow ceases. The terrain-following theory is compared to a standard depth-averaged avalanche model in an aligned Cartesian coordinate system. For this very sensitive problem, it is shown that the steady-shock regime is captured significantly better by the terrain-following avalanche model, and that the standard theory is unable to predict the take-off point of the jet. To retain the practical simplicity of using Cartesian coordinates, but have the improved predictive power of the terrain-following model, a coordinate mapping is used to transform the terrain-following equations from curvilinear to Cartesian coordinates. The terrain-following model, in Cartesian coordinates, makes identical predictions to the original curvilinear formulation, but is much simpler to implement.

scholarly journals The fate of random initial vorticity distributions for two-dimensional Euler equations on a sphere

2015 ◽  
Vol 786 ◽  
pp. 1-4 ◽  
Author(s):  
Paul K. Newton
Keyword(s):  
Numerical Simulations ◽  
Euler Equations ◽  
Large Scale ◽  
Initial Conditions ◽  
Infinite Family ◽  
Small Scale ◽  
Two Dimensional ◽  
Random Initial Conditions ◽  
Long Time ◽  
Initial Vorticity

The paper by Dritschel et al. (J. Fluid Mech., vol. 783, 2015, pp. 1–22) describes the long-time behaviour of inviscid two-dimensional fluid dynamics on the surface of a sphere. At issue is whether the flow settles down to an equilibrium or whether, for generic (random) initial conditions, the long-time solution is periodic, quasi-periodic or chaotic. While it might be surprising that this issue is not settled in the literature, it is important to keep in mind that the Euler equations form a dissipationless Hamiltonian system, hence the set of equations only redistributes the initial vorticity, generating smaller and smaller scales, while keeping kinetic energy, angular impulse and an infinite family of vorticity moments (Casimirs) intact. While special solutions that never settle down to an equilibrium state can be constructed using point vortices, vortex patches and other distributions, the fate of random initial conditions is a trickier problem. Previous statistical theories indicate that the long-time state should be a stationary large-scale distribution of vorticity. By carrying out careful numerical simulations using two different methods, the authors make a compelling case that the generic long-time state resembles a large-scale oscillating quadrupolar vorticity field, surrounded by persistent small-scale vortices. While numerical simulations can never conclusively settle this issue, the results might help guide future theories that seek to prove the existence of such an interesting dynamical long-time state.

Representation of Reservoir Geometry for Numerical Simulation

1970 ◽  
Vol 10 (04) ◽  
pp. 393-404 ◽  
Author(s):  
G.J. Hirasaki ◽  
P.M. O'Dell
Keyword(s):  
Coordinate System ◽  
Curvilinear Coordinates ◽  
Reference Plane ◽  
Cartesian Coordinate System ◽  
Curvilinear Coordinate System ◽  
Coordinate Systems ◽  
Cartesian Coordinates ◽  
Curvilinear Coordinate ◽  
Reservoir Geometry ◽  
Cartesian Coordinate

Abstract For most reservoirs the reservoir thickness and dip vary with position. For such reservoirs, the use of a Cartesian coordinate system is awkward as the coordinate surfaces are planes and the finite-difference grid elements are rectangular parallepipeds. However, these reservoirs may be efficiently parallepipeds. However, these reservoirs may be efficiently modeled with a curvilinear coordinate system that has coordinate surfaces that coincide with the reservoir surfaces. A procedure is presented that may be used to determine a curvilinear coordinate system that will conform with the geometry of the reservoir. The reservoir geometry is described by the depth of the top of the reservoir and the thickness. The mass conservation equations are presented in curvilinear coordinates. The finite-difference equations differ from the usual Cartesian coordinate formulation by a factor multiplying the pore volume and transmissibilities. A numerical example is presented to illustrate the magnitude of the error that may occur in the computed oil recovery if the Cartesian coordinate system is simply modified to yield the correct depth and pore volumes. Introduction Many reservoirs have a shape that is inconvenient and possibly inaccurate to model with Cartesian coordinates. The use of a curvilinear coordinate system that follows the shape of the reservoir can be advantageous for such reservoirs. The formulation discussed here will have the greatest advantage in modeling thin reservoirs but will have little advantage in modeling a reservoir whose thickness is greater than its radius of curvature, such as a pinnacle reef. pinnacle reef. In this paper the reader is introduced to various grid systems used to model reservoirs. A brief discussion of some concepts of differential geometry contrasts differences between Cartesian coordinates and curvilinear coordinates. A curvilinear coordinate system for modeling reservoir geometry is presented. Formulation of the conservation equations in curvilinear coordinates and the necessary modifications to pore volume and transmissibility are discussed. A numerical example illustrates the magnitude of the error that may result from some coordinate systems. COORDINATE SYSTEMS AND RESERVOIR GRID NETWORKS A reservoir is usually described with the depth, thickness, boundaries, etc., shown on a structure map with sea level as a reference plane. For example, the subsea depth may be shown as a contour map on the reference plane with a Cartesian coordinate grid superimposed on the reference plane as shown on Fig. 1. The Cartesian coordinates, plane as shown on Fig. 1. The Cartesian coordinates, (y1, y2), have been defined as the coordinates for the reference plane. If the reservoir surfaces are parallel planes, Cartesian coordinates may be used. The Cartesian coordinate may be rotated such that the coordinate surfaces coincide with the reservoir surfaces. SPEJ P. 393

Gradient enhancement and filament ejection for a non-uniform elliptic vortex in two-dimensional turbulence

2001 ◽  
Vol 439 ◽  
pp. 43-56 ◽  
Author(s):  
Y. KIMURA ◽  
J. R. HERRING
Keyword(s):  
Compact Support ◽  
Initial Conditions ◽  
Three Dimensions ◽  
Energy Spectra ◽  
Small Scale ◽  
Strain Rate Tensor ◽  
Two Dimensional ◽  
Similar Form ◽  
Ejection Process ◽  
Gradient Enhancement

The axisymmetrization of a two-dimensional non-uniform elliptic vortex is studied in terms of the growth of palinstrophy, the squared vorticity gradient. First, it is pointed out that the equation for palinstrophy growth, if written in terms of the strain rate tensor, has a similar form to that of enstrophy growth in three-dimensions – the vortex-stretching equation. Then palinstrophy production is analysed, particularly for non-uniform elliptic vortices. It is shown analytically and verified numerically that a non-uniform elliptic vortex in general has a quadrupole structure for palinstrophy production, and that in the positive production regions, vortex filaments are ejected following the gradient enhancement process for vorticity. Numerical simulations are conducted for two different initial conditions, compact support and Gaussian vorticity distributions. These are characterized by distinctly different features of filament ejection and energy spectra. For both cases, the total palinstrophy production is a good indicator of the development of small-scale vorticity. In particular for the compact support case, a possible intermittency mechanism in the filament ejection process is proposed.

scholarly journals A two-dimensional glacier–fjord coupled model applied to estimate submarine melt rates and front position changes of Hansbreen, Svalbard

2018 ◽  
Vol 64 (247) ◽  
pp. 745-758 ◽  
Author(s):  
E. DE ANDRÉS ◽  
J. OTERO ◽  
F. NAVARRO ◽  
A. PROMIŃSKA ◽  
J. LAPAZARAN ◽  
...  
Keyword(s):  
Coupled Model ◽  
Small Scale ◽  
Two Dimensional ◽  
Time Step ◽  
Software Packages ◽  
Front Position ◽  
Circulation Patterns ◽  
Glacier Front ◽  
Order Of Magnitude ◽  
Frontal Ablation

ABSTRACTWe have developed a two-dimensional coupled glacier–fjord model, which runs automatically using Elmer/Ice and MITgcm software packages, to investigate the magnitude of submarine melting along a vertical glacier front and its potential influence on glacier calving and front position changes. We apply this model to simulate the Hansbreen glacier–Hansbukta proglacial–fjord system, Southwestern Svalbard, during the summer of 2010. The limited size of this system allows us to resolve some of the small-scale processes occurring at the ice–ocean interface in the fjord model, using a 0.5 s time step and a 1 m grid resolution near the glacier front. We use a rich set of field data spanning the period April–August 2010 to constrain, calibrate and validate the model. We adjust circulation patterns in the fjord by tuning subglacial discharge inputs that best match observed temperature while maintaining a compromise with observed salinity, suggesting a convectively driven circulation in Hansbukta. The results of our model simulations suggest that both submarine melting and crevasse hydrofracturing exert important controls on seasonal frontal ablation, with submarine melting alone not being sufficient for reproducing the observed patterns of seasonal retreat. Both submarine melt and calving rates accumulated along the entire simulation period are of the same order of magnitude, ~100 m. The model results also indicate that changes in submarine melting lag meltwater production by 4–5 weeks, which suggests that it may take up to a month for meltwater to traverse the englacial and subglacial drainage network.

MAGNETIC FIELD AMPLIFICATION BY RELATIVISTIC SHOCKS IN AN INHOMOGENEOUS MEDIUM

2012 ◽  
Vol 08 ◽  
pp. 364-367
Author(s):  
YOSUKE MIZUNO ◽  
MARTIN POHL ◽  
JACEK NIEMIEC ◽  
BING ZHANG ◽  
KEN-ICHI NISHIKAWA ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Inhomogeneous Medium ◽  
Magnetic Energy ◽  
Small Scale ◽  
Two Dimensional ◽  
Filamentary Structure ◽  
Field Amplification ◽  
Relativistic Shocks ◽  
Field Lines ◽  
Magnetohydrodynamic Simulations

We perform two-dimensional relativistic magnetohydrodynamic simulations of a mildly relativistic shock propagating through an inhomogeneous medium. We show that the postshock region becomes turbulent owing to preshock density inhomogeneity, and the magnetic field is strongly amplified due to the stretching and folding of field lines in the turbulent velocity field. The amplified magnetic field evolves into a filamentary structure in two-dimensional simulations. The magnetic energy spectrum is flatter than the Kolmogorov spectrum and indicates that the so-called small-scale dynamo is occurring in the postshock region. We also find that the amplitude of magnetic-field amplification depends on the direction of the mean preshock magnetic field.

scholarly journals Saturation of Zeldovich stretch–twist–fold map dynamos

2015 ◽  
Vol 81 (5) ◽  
Author(s):  
Amit Seta ◽  
Pallavi Bhat ◽  
Kandaswamy Subramanian
Keyword(s):  
Initial Conditions ◽  
Reynolds Numbers ◽  
Small Scale ◽  
Saturation Properties ◽  
Field Amplification ◽  
Saturated State ◽  
Shape Invariant ◽  
Fold Map ◽  
Scale Fluctuation ◽  
The Magnetic Field

Zeldovich’s stretch–twist–fold (STF) dynamo provided a breakthrough in conceptual understanding of fast dynamos, including the small-scale fluctuation dynamos. We study the evolution and saturation behaviour of two types of generalized Baker’s map dynamos, which have been used to model Zeldovich’s STF dynamo process. Using such maps allows one to analyse dynamos at much higher magnetic Reynolds numbers $\mathit{Re}_{M}$ as compared to direct numerical simulations. In the two-strip map dynamo there is constant constructive folding, while the four-strip map dynamo also allows the possibility of a destructive reversal of the field. Incorporating a diffusive step parametrized by $\mathit{Re}_{M}$ into the map, we find that the magnetic field $B(x)$ is amplified only above a critical $\mathit{Re}_{M}=R_{\mathit{crit}}\sim 4$ for both types of dynamos. The growing $B(x)$ approaches a shape-invariant eigenfunction independent of initial conditions, whose fine structure increases with increasing $\mathit{Re}_{M}$. Its power spectrum $M(k)$ displays sharp peaks reflecting the fractal nature of $B(x)$ above the diffusive scale. We explore the saturation of these dynamos in three ways: via a renormalized reduced effective $\mathit{Re}_{M}$ (case I) or due to a decrease in the efficiency of the field amplification by stretching, without changing the map (case IIa), or changing the map (case IIb), and a combination of both effects (case III). For case I, we show that $B(x)$ in the saturated state, for both types of maps, approaches the marginal eigenfunction, which is obtained for $\mathit{Re}_{M}=R_{\mathit{crit}}$ independent of the initial $\mathit{Re}_{M}=R_{M0}$. On the other hand, in case II, for the two-strip map, we show that $B(x)$ saturates, preserving the structure of the kinematic eigenfunction. Thus the energy is transferred to larger scales in case I but remains at the smallest resistive scales in case II, as can be seen from both $B(x)$ and $M(k)$. For the four-strip map, $B(x)$ oscillates with time, although with a structure similar to the kinematic eigenfunction. Interestingly, the saturated state in case III shows an intermediate behaviour, with $B(x)$ similar to the kinematic eigenfunction at an intermediate $\mathit{Re}_{M}=R_{\mathit{sat}}$, with $R_{M0}>R_{\mathit{sat}}>R_{\mathit{crit}}$. The $R_{\mathit{sat}}$ value is determined by the relative importance of the increased diffusion versus the reduced stretching. These saturation properties are akin to the range of possibilities that have been discussed in the context of fluctuation dynamos.

Experiments on wind-driven heat exchange processes over melting snow

2021 ◽  
Author(s):  
Michael Haugeneder ◽  
Tobias Jonas ◽  
Dylan Reynolds ◽  
Michael Lehning ◽  
Rebecca Mott
Keyword(s):  
Snow Cover ◽  
Surface Wind ◽  
Infrared Camera ◽  
Snow Accumulation ◽  
Internal Boundary ◽  
Small Scale ◽  
Snow Melt ◽  
Two Dimensional ◽  
Atmospheric Conditions ◽  
Near Surface

<p>Snowmelt runoff predictions in alpine catchments are challenging because of the high spatial variability of t<span>he snow cover driven by </span>various snow accumulation and ablation processes. In spring, the coexistence of bare and snow-covered ground engages a number of processes such as the enhanced lateral advection of heat over partial snow cover, the development of internal boundary layers, and atmospheric decoupling effects due to increasing stability at the snow cover. The interdependency of atmospheric conditions, topographic settings and snow coverage remains a challenge to accurately account for these processes in snow melt models.<br>In this experimental study, we used an Infrared Camera (VarioCam) pointing at thin synthetic projection screens with negligible heat capacity. Using the surface temperature of the screen as a proxy for the air temperature, we obtained a two-dimensional instantaneous measurement. Screens were installed across the transition between snow-free and snow-covered areas. With IR-measurements taken at 10Hz, we capture<span> the dynamics of turbulent temperature fluctuations</span><span> </span>over the patchy snow cover at high spatial and temporal resolution. From this data we were able to obtain high-frequency, two-dimensional windfield estimations adjacent to the surface.</p><p>Preliminary results show the formation of a stable internal boundary layer (SIBL), which was temporally highly variable. Our data suggest that the SIBL height is very shallow and strongly sensitive to the mean near-surface wind speed. Only strong gusts were capable of penetrating through this SIBL leading to an enhanced energy input to the snow surface.</p><p>With these type of results from our experiments and further measurements this spring we aim to better understand small scale energy transfer processes over patch snow cover and it’s dependency on the atmospheric conditions, enabling to improve parameterizations of these processes in coarser-resolution snow melt models.</p>

Microdeformations in Sands by Digital Image Processing and Analysis

1996 ◽  
Vol 1548 (1) ◽  
pp. 31-37 ◽  
Author(s):  
Scott A. Raschke ◽  
Roman D. Hryciw ◽  
Gregory W. Donohoe
Keyword(s):  
Image Processing ◽  
Digital Image Processing ◽  
Digital Image ◽  
Small Scale ◽  
Two Dimensional ◽  
Localized Deformation ◽  
Displacement Fields ◽  
Particulate Media ◽  
Image Processing And Analysis ◽  
Video Images

Laboratory experiments are typically performed on particulate media to study stress-deformation behavior and to verify or calibrate computer models from controlled or measured boundary stresses and displacements. However, such data do not permit the formation of shear bands, displacement fields within flowing granular media, and other small-scale localized deformation phenomena to be identified. Described are two semiautomated computer vision techniques for accurately determining the two-dimensional displacement field in granular soils from video images obtained through a transparent planar viewing window. The techniques described are applicable for studying the behavior of particulate media under plane strain and certain axisymmetric test conditions. Digital image processing and analysis routines are used in two different computer programs, Tracker and Tracer, Tracker uses a graphical user interface that allows individual particles to be selected and tracked through a sequence of digital video images. A contrast edge detection algorithm delineates the two-dimensional projected boundaries of particles. The location of the centroid of each particle selected for tracking is determined from the boundary to quantify the trajectory of each particle. Tracer maps the trace or trajectory of specially dyed fluorescent particles in a sequence of video frames. A thresholding technique segments individual particle trajectories. Together, Tracker and Tracer provide a set of tools for identifying small-scale displacement fields in particulate assemblies deforming under either quasi-static or rapid loading (such as gravity flow).

scholarly journals Toward understanding the multiscale spatial spectrum of solar convection

2012 ◽  
Vol 8 (S294) ◽  
pp. 361-363
Author(s):  
A. V. Getling ◽  
O. S. Mazhorova ◽  
O. V. Shcheritsa
Keyword(s):  
Convective Instability ◽  
Convective Flow ◽  
Large Scale ◽  
Fluid Layer ◽  
Boussinesq Equations ◽  
Spatial Spectrum ◽  
Small Scale ◽  
Two Dimensional ◽  
Solar Convection

AbstractConvection is simulated numerically based on two-dimensional Boussinesq equations for a fluid layer with a specially chosen stratification such that the convective instability is much stronger in a thin subsurface sublayer than in the remaining part of the layer. The developing convective flow has a small-scale component superposed onto a basic large-scale roll flow.

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