scholarly journals Energy considerations in accelerating rapid shear granular flows

2009 ◽  
Vol 16 (3) ◽  
pp. 399-407 ◽  
Author(s):  
S. P. Pudasaini ◽  
B. Domnik
Keyword(s):  
Total Energy ◽  
Frictional Resistance ◽  
Stationary Flow ◽  
Momentum Conservation ◽  
Stationary Flows ◽  
Surface Gradient ◽  
Bulk Motion ◽  
Order Of Magnitude ◽  
Friction Energy ◽  
Gravity Acceleration

Abstract. We present a complete expression for the total energy associated with a rapid frictional granular shear flow down an inclined surface. This expression reduces to the often used energy for a non-accelerating flow of an isotropic, ideal fluid in a horizontal channel, or to the energy for a vertically falling mass. We utilize thickness-averaged mass and momentum conservation laws written in a slope-defined coordinate system. Both the enhanced gravity and friction are taken into account in addition to the bulk motion and deformation. The total energy of the flow at a given spatial position and time is defined as the sum of four energy components: the kinetic energy, gravity, pressure and the friction energy. Total energy is conserved for stationary flow, but for non-stationary flow the non-conservative force induced by the free-surface gradient means that energy is not conserved. Simulations and experimental results are used to sketch the total energy of non-stationary flows. Comparison between the total energy and the sum of the kinetic and pressure energy shows that the contribution due to gravity acceleration and frictional resistance can be of the same order of magnitude, and that the geometric deformation plays an important role in the total energy budget of the cascading mass. Relative importance of the different constituents in the total energy expression is explored. We also introduce an extended Froude number that takes into account the apparent potential energy induced by gravity and pressure.

Conservation laws

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Dynamical System ◽  
Angular Momentum ◽  
Conservation Laws ◽  
Total Energy ◽  
Superposition Principle ◽  
Momentum Conservation ◽  
Conserved Quantities ◽  
Angular Momentum Conservation ◽  
Third Law ◽  
The Law

This chapter defines the conserved quantities associated with an isolated dynamical system, that is, the quantities which remain constant during the motion of the system. The law of momentum conservation follows directly from Newton’s third law. The superposition principle for forces allows Newton’s law of motion for a body Pa acted on by other bodies Pa′ in an inertial Cartesian frame S. The law of angular momentum conservation holds if the forces acting on the elements of the system depend only on the separation of the elements. Finally, the conservation of total energy requires in addition that the forces be derivable from a potential.

scholarly journals Powder flow down a vertical pipe: the effect of air flow

2002 ◽  
Vol 459 ◽  
pp. 317-345 ◽  
Author(s):  
Y. BERTHO ◽  
F. GIORGIUTTI-DAUPHINÉ ◽  
T. RAAFAT ◽  
E. J. HINCH ◽  
H. J. HERRMANN ◽  
...  
Keyword(s):  
Small Diameter ◽  
Stationary Flow ◽  
Momentum Conservation ◽  
Wall Friction ◽  
Particle Fraction ◽  
Friction Forces ◽  
Flow Rates ◽  
Lower Flow ◽  
Grain Flow ◽  
Grain Distribution

The dynamics of dry granular flows down a vertical glass pipe of small diameter have been studied experimentally. Simultaneous measurements of pressure profiles, air and grain flow rates and volume fractions of particles have been realized together with spatio-temporal diagrams of the grain distribution down the tube. At large grain flow rates, one observes a stationary flow characterized by high particle velocities, low particle fractions and a downflow of air resulting in an underpressure in the upper part of the pipe. A simple model assuming a free fall of the particles slowed down by air friction and taking into account finite particle fraction effects through Richardson–Zaki's law has been developed: it reproduces pressure and particle fraction variations with distance and estimates friction forces with the wall. At lower flow rates, sequences of high-density plugs separated by low-density bubbles moving down at a constant velocity are observed. The pressure is larger than outside the tube and its gradient reflects closely the weight of the grains. Writing mass and momentum conservation equations for the air and for the grains allows one to estimate the wall friction, which is less than 10% of the weight for grains with a clean smooth surface but up to 30% for grains with a rougher surface. At lower flow rates, oscillating-wave regimes resulting in large pressure fluctuations are observed and their frequency is predicted.

Fluctuation of cilia-generated flow on the surface of the tracheal lumen

2014 ◽  
Vol 306 (2) ◽  
pp. L144-L151 ◽  
Author(s):  
Kouki Kiyota ◽  
Hironori Ueno ◽  
Keiko Numayama-Tsuruta ◽  
Tomofumi Haga ◽  
Yohsuke Imai ◽  
...  
Keyword(s):  
Flow Field ◽  
Transport Phenomena ◽  
Spatial Inhomogeneity ◽  
Tracheal Lumen ◽  
Ciliated Cells ◽  
Flow Fluctuations ◽  
Flow Fluctuation ◽  
Bulk Motion ◽  
Order Of Magnitude ◽  
Directional Transport

Although we inhale air that contains many harmful substances, including, for example, dust and viruses, these small particles are trapped on the surface of the tracheal lumen and transported towards the larynx by cilia-generated flow. The transport phenomena are affected not only by the time- and space-average flow field but also by the fluctuation of the flow. Because flow fluctuation has received little attention, we investigated it experimentally in mice. To understand the origin of flow fluctuation, we first measured the distribution of ciliated cells in the trachea and individual ciliary motions. We then measured the detailed flow field using a confocal micro-PTV system. Strong flow fluctuations were observed, caused by the unsteadiness of the ciliary beat and the spatial inhomogeneity of ciliated cells. The spreading of particles relative to the bulk motion became diffusive if the time scale was sufficiently larger than the beat period. Finally, we quantified the effects of flow fluctuation on bulk flow by evaluating the Peclet number of the system, which indicated that the directional transport was an order of magnitude larger than the isotropic diffusion. These results are important in understanding transport phenomena in the airways on a cellular scale.

scholarly journals Non-Empirical BEM Corrections Relating to Angular and Axial Momentum Conservation

Energies ◽  
2019 ◽  
Vol 12 (2) ◽  
pp. 320
Author(s):  
Søren Hjort
Keyword(s):  
Power Efficiency ◽  
Correction Term ◽  
Outer Part ◽  
Momentum Conservation ◽  
Navier Stokes ◽  
Speed Ratio ◽  
Power Extraction ◽  
Higher Power ◽  
Horizontal Axis Wind Turbines ◽  
Order Of Magnitude

The Blade-Element Momentum (BEM) model for Horizontal-Axis Wind Turbines (HAWTs), although extremely useful, is known to be approximate due to model formulation insufficiencies, for which add-ons and corrections have been formulated over the past many decades. Scrutiny of the axial and azimuthal momentum conservation properties reveals momentum simplifications and absence of momentum sources not included in momentum theory underlying the standard BEM. One aspect relates to azimuthal momentum conservation, the wake swirl. This correction can be expressed analytically. Another aspect relates to axial momentum conservation, the wake expansion. This correction is not analytically quantifiable. The latter correction term is therefore quantified from postprocessing a large number of axisymmetric Actuator Disk (AD) Navier-Stokes computations with systematic variation of disk loading and tip-speed ratio. The new momentum correction terms are then included in the BEM model, and results benchmarked against references. The corrected BEM is derived by re-visiting the governing equations. For a disk represented by a constant-circulation set of blades, the corrected BEM contains no approximation to the underlying conservation laws. The study contributes by bridging the gap between BEM and the axisymmetric AD method for all disk load levels and tip speed ratios relevant for a wind turbine. The wake swirl correction leads to higher power efficiency at lower tip-speed ratios. The wake expansion correction causes a redistribution of the potential for power extraction, which increases on the inner part of the rotor and decreases on the outer part of the rotor. The overall rotor-averaged value of Betz limit is unaffected by the new corrections, but exceeding Betz locally on the inner- and mid-section of the rotor is shown to be possible. The two corrections significantly improve the axi-symmetric static BEM modelling accuracy for the radial distributions as well as for the rotor-integrated quantities, by reducing errors, approximately one order of magnitude. The relevance of these corrections for modern multi-MW rotors is quantified and discussed.

scholarly journals The problem of the second wind turbine – a note on a common but flawed wind power estimation method

2012 ◽  
Vol 3 (1) ◽  
pp. 79-86 ◽  
Author(s):  
F. Gans ◽  
L. M. Miller ◽  
A. Kleidon
Keyword(s):  
Boundary Layer ◽  
Simple Model ◽  
Kinetic Energy ◽  
Atmospheric Boundary Layer ◽  
Wind Turbine ◽  
Wind Power ◽  
Momentum Conservation ◽  
Height Velocity ◽  
Power Extraction ◽  
Order Of Magnitude

Abstract. Several recent wind power estimates suggest that this renewable energy resource can meet all of the current and future global energy demand with little impact on the atmosphere. These estimates are calculated using observed wind speeds in combination with specifications of wind turbine size and density to quantify the extractable wind power. However, this approach neglects the effects of momentum extraction by the turbines on the atmospheric flow that would have effects outside the turbine wake. Here we show with a simple momentum balance model of the atmospheric boundary layer that this common methodology to derive wind power potentials requires unrealistically high increases in the generation of kinetic energy by the atmosphere. This increase by an order of magnitude is needed to ensure momentum conservation in the atmospheric boundary layer. In the context of this simple model, we then compare the effect of three different assumptions regarding the boundary conditions at the top of the boundary layer, with prescribed hub height velocity, momentum transport, or kinetic energy transfer into the boundary layer. We then use simulations with an atmospheric general circulation model that explicitly simulate generation of kinetic energy with momentum conservation. These simulations show that the assumption of prescribed momentum import into the atmospheric boundary layer yields the most realistic behavior of the simple model, while the assumption of prescribed hub height velocity can clearly be disregarded. We also show that the assumptions yield similar estimates for extracted wind power when less than 10% of the kinetic energy flux in the boundary layer is extracted by the turbines. We conclude that the common method significantly overestimates wind power potentials by an order of magnitude in the limit of high wind power extraction. Ultimately, environmental constraints set the upper limit on wind power potential at larger scales rather than detailed engineering specifications of wind turbine design and placement.

scholarly journals Dynamically consistent entrainment laws for depth-averaged avalanche models

2014 ◽  
Vol 759 ◽  
pp. 701-738 ◽  
Author(s):  
Dieter Issler
Keyword(s):  
Stationary Flow ◽  
Analytic Solutions ◽  
Rheological Parameters ◽  
Bottom Surface ◽  
Entrainment Rate ◽  
Stationary Flows ◽  
Start Up ◽  
Flow Variables ◽  
Bed Material ◽  
Parameter Values

AbstractThe bed entrainment rate in a gravity mass flow (GMF) is uniquely determined by the properties of the bed and the flow. In depth-averaging, however, critical information on the flow variables near the bed is lost and empirical assumptions usually are made instead. We study the interplay between bed and flow assuming a perfectly brittle bed, characterized by its shear strength ${\it\tau}_{c}$, and erosion along the bottom surface of the flow; frontal entrainment is neglected here. The brittleness assumption implies that the shear stress at the bed surface cannot exceed ${\it\tau}_{c}$. For quasi-stationary flows in a simplified setting, analytic solutions are found for Bingham and frictional–collisional (FC) fluids. Extending this theory to non-stationary flows requires some assumptions for the velocity profile. For the Bingham fluid, the profile of a ‘proxy’ quasi-stationary eroding flow is used; the rheological parameters are chosen to match the instantaneous velocity and shear-layer depth of the non-stationary flow. For the FC fluid, a two-parameter family of functions that closely match the profiles obtained in depth-resolved numerical simulations is assumed; the boundary conditions determine the instantaneous parameter values and allow computation of the erosion rate. Preliminary tests with the FC erosion formula incorporated in a simple slab model indicate that the non-stationary erosion formula matches the depth-resolved simulations asymptotically, but differs in the start-up phase. The non-stationary erosion formulae are valid only up to a limit velocity (and to a limit flow depth if there is Coulomb friction). This appears to mark the transition to another erosion regime – to be described by a different model – where chunks of bed material are intermittently ripped out and gradually entrained into the flow.

Velocity and Momentum Decay Characteristics of a Submerged Viscoplastic Jet

2013 ◽  
Vol 136 (2) ◽  
Author(s):  
Khaled J. Hammad
Keyword(s):  
Flow Field ◽  
Yield Stress ◽  
Initial Velocity ◽  
Momentum Conservation ◽  
Wide Range ◽  
Velocity Decay ◽  
Flow Field Characteristics ◽  
Order Of Magnitude ◽  
Field Information ◽  
Penetration Depths

Velocity and momentum decay characteristics of a submerged viscoplastic non-Newtonian jet are studied within the steady laminar flow regime. The governing mass and momentum conservation equations along with the Bingham rheological model are solved numerically using a finite-difference scheme. A parametric study is performed to reveal the influence of the initial velocity profile, flow inertia, and yield stress presence on the flow field characteristics. Two initial velocity profiles are considered, a top-hat and fully developed pipe jets. The centerline velocity decay is found to be more rapid for the pipe jet than the top-hat one when the fluid is Newtonian while the opposite trend is observed for yield stress Bingham fluids. The decay in the momentum flux of the pipe jet is always less than that of the top-hat jet. Momentum and velocity based jet depths of penetration are introduced and used to analyze the obtained flow field information for a wide range of Reynolds and yield numbers. Depths of penetration are found to linearly increase with the Reynolds number and substantially decrease with the yield number. The presence of yield stress significantly reduces the momentum and velocity penetration depths of submerged top-hat and pipe jets. Penetration depths of yield stress fluids are shown to be more than an order of magnitude smaller than the ones corresponding to Newtonian fluids.

Nonlinear waves propagating transverse to the magnetic field

2001 ◽  
Vol 65 (3) ◽  
pp. 213-233 ◽  
Author(s):  
J. F. McKENZIE ◽  
E. DUBININ ◽  
K. SAUER
Keyword(s):  
Magnetic Field ◽  
Mach Number ◽  
Soliton Solutions ◽  
Momentum Conservation ◽  
Plasma Pressure ◽  
Mirror Image ◽  
Stationary Flows ◽  
Classical Work ◽  
Mach Numbers ◽  
The Magnetic Field

We generalize the classical work of Adlam and Allen [Phil. Mag.3, 448 (1958)] on solitons in a cold plasma propagating perpendicular to the magnetic field to include the effects of plasma pressure. This is done by making extensive use of the properties of total momentum conservation (denoted by the term ‘momentum hodograph’, since it yields a locus in the plane of the electron and proton speeds in the direction of the wave) and the energy integral of the system as a whole. These relations elucidate the phase and integral curves of stationary flows, from which soliton solutions may be constructed. In general, only compressive solitons are permitted, and we have found an analytical expression for the critical fast Mach number as a function of the proton acoustic Mach number, which shows that it varies from its classical value of 2 (at large proton acoustic Mach numbers) to unity, where the incoming flow is proton-sonic. At the critical fast Mach number, two possible soliton-like solutions can be constructed. One is the classical compression, in which the magnetic field develops a cusp in the centre of the wave. The other is a compression in the magnetic field followed by a deep depression in the centre of the wave, which is completed by the mirror image of this signature of compression–rarefaction. This structure involves a smooth supersonic–subsonic transition in the proton flow. For Mach numbers in excess of the critical one, this kind of structure can also be constructed, but now the magnetic field is cusp-like at the points of maximum compression.

scholarly journals Dynamics and mechanics of tracer particles

2014 ◽  
Vol 2 (1) ◽  
pp. 429-476 ◽  
Author(s):  
C. B. Phillips ◽  
D. J. Jerolmack
Keyword(s):  
Tracer Particle ◽  
Shear Velocity ◽  
Momentum Conservation ◽  
Bed Load ◽  
Modelling Framework ◽  
River Bed ◽  
Tracer Particles ◽  
Spatial Sorting ◽  
Order Of Magnitude ◽  
The Individual

Abstract. Understanding the mechanics of bed load at the flood scale is necessary to link hydrology to landscape evolution. Here we report on observations of the transport of coarse sediment tracer particles in a cobble-bedded alluvial river and a step-pool tributary, at the individual flood and multi-annual timescales. Tracer particle data for each survey are composed of measured displacement lengths for individual particles, and the number of tagged particles mobilized. For single floods we find that: measured tracer particle displacement lengths are exponentially distributed; the number of mobile particles increases linearly with peak flood Shields stress, indicating partial bed load transport for all observed floods; and modal displacement lengths scale linearly with excess shear velocity. These findings provide quantitative field support for a recently proposed modelling framework based on momentum conservation at the grain scale. Tracer displacement shows a weak correlation with particle size at the individual flood scale, however cumulative travel distance begins to show an inverse relation to grain size when measured over many transport events. The observed spatial sorting of tracers approaches that of the river bed, and is consistent with size-selective deposition models and laboratory experiments. Tracer displacement data for the step-pool and alluvial channels collapse onto a single curve – despite more than an order of magnitude difference in channel slope – when variations of critical Shields stress and flow resistance between the two are accounted for. Results show how bed load dynamics may be predicted from a record of river stage, providing a direct link between climate and sediment transport.

Depth of Penetration of a Submerged Viscoplastic Non-Newtonian Jet

2012 ◽  
Author(s):  
Khaled J. Hammad
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Flow Field ◽  
Rheological Model ◽  
Momentum Conservation ◽  
Depth Of Penetration ◽  
Wide Range ◽  
Order Of Magnitude ◽  
Field Information ◽  
Penetration Depths

Depth of penetration characteristics of a submerged viscoplastic non-Newtonian jet were studied by numerically solving the governing mass and momentum conservation equations along with the Bingham rheological model. Momentum and velocity based jet depths of penetration were introduced and used to analyze the obtained steady and laminar flow field information for a wide range of Reynolds and yield numbers. Depths of penetration were found to linearly increase with the Reynolds number and substantially decrease with the yield number. Penetration depths of yield stress fluids were shown to be more than an order of magnitude smaller than the ones corresponding to Newtonian fluids.

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