scholarly journals Hillslope-storage Boussinesq model for subsurface flow and variable source areas along complex hillslopes: 2. Intercomparison with a three-dimensional Richards equation model

2003 ◽  
Vol 39 (11) ◽  
Author(s):  
Claudio Paniconi ◽  
Peter A. Troch ◽  
E. Emiel van Loon ◽  
Arno G. J. Hilberts
Keyword(s):  
Subsurface Flow ◽  
Three Dimensional ◽  
Equation Model ◽  
Richards Equation ◽  
Boussinesq Model ◽  
Variable Source ◽  
Source Areas

Laminar natural convection about an isothermally heated sphere at small Grashof number

1968 ◽  
Vol 34 (1) ◽  
pp. 163-176 ◽  
Author(s):  
Francis E. Fendell
Keyword(s):  
Natural Convection ◽  
Three Dimensional ◽  
Perturbation Expansion ◽  
Outer Sphere ◽  
Convective Transport ◽  
Convection Flow ◽  
Constant Density ◽  
Boussinesq Model ◽  
Grashof Number ◽  
Recirculating Flow

The flow induced by gravity about a very small heated isothermal sphere introduced into a fluid in hydrostatic equilibrium is studied. The natural-convection flow is taken to be steady and laminar. The conditions under which the Boussinesq model is a good approximation to the full conservation laws are described. For a concentric finite cold outer sphere with radius, in ratio to the heated sphere radius, roughly less than the Grashof number to the minus one-half power, a recirculating flow occurs; fluid rises near the inner sphere and falls near the outer sphere. For a small heated sphere in an unbounded medium an ordinary perturbation expansion essentially in the Grashof number leads to unbounded velocities far from the sphere; this singularity is the natural-convection analogue of the Whitehead paradox arising in three-dimensional low-Reynolds-number forced-convection flows. Inner-and-outer matched asymptotic expansions reveal the importance of convective transport away from the sphere, although diffusive transport is dominant near the sphere. Approximate solution is given to the nonlinear outer equations, first by seeking a similarity solution (in paraboloidal co-ordinates) for a point heat source valid far from the point source, and then by linearization in the manner of Oseen. The Oseen solution is matched to the inner diffusive solution. Both outer solutions describe a paraboloidal wake above the sphere within which the enthalpy decays slowly relative to the rapid decay outside the wake. The updraft above the sphere is reduced from unbounded growth with distance from the sphere to constant magnitude by restoration of the convective accelerations. Finally, the role of vertical stratification of the ambient density in eventually stagnating updrafts predicted on the basis of a constant-density atmosphere is discussed.

A new numerical method to approximate root water uptake fluxes in a mixed-dimensional 1D-3D setting

2021 ◽  
Author(s):  
Timo Koch ◽  
Hanchuan Wu ◽  
Kent-André Mardal ◽  
Rainer Helmig ◽  
Martin Schneider
Keyword(s):  
Analytical Solutions ◽  
Water Pressure ◽  
Numerical Test ◽  
Three Dimensional ◽  
Root Water Uptake ◽  
Richards Equation ◽  
Soil Pressure ◽  
Approximation Accuracy ◽  
Flux Approximation ◽  
Local Water

<p>1D-3D methods are used to describe root water and nutrient uptake in complex root networks. Root systems are described as networks of line segments embedded in a three-dimensional soil domain. Particularly for dry soils, local water pressure and nutrient concentration gradients can be become very large in the vicinity of roots. Commonly used discretization lengths (for example 1cm) in root-soil interaction models do not allow to capture these gradients accurately. We present a new numerical scheme for approximating root-soil interface fluxes. The scheme is formulated in the continuous PDE setting so that is it formally independent of the spatial discretization scheme (e.g. FVM, FD, FEM). The interface flux approximation is based on a reconstruction of interface quantities using local analytical solutions of the steady-rate Richards equation. The local mass exchange is numerically distributed in the vicinity of the root. The distribution results in a regularization of the soil pressure solution which is easier to approximate numerically. This technique allows for coarser grid resolutions while maintaining approximation accuracy. The new scheme is verified numerically against analytical solutions for simplified cases. We also explore limitations and possible errors in the flux approximation with numerical test cases. Finally, we present the results of a recently published benchmark case using this new method.</p>

scholarly journals The Yamada Model for a Self-Pulsing Laser: Bifurcation Structure for Nonidentical Decay Times of Gain and Absorber

2020 ◽  
Vol 30 (14) ◽  
pp. 2030039
Author(s):  
Robert Otupiri ◽  
Bernd Krauskopf ◽  
Neil G. R. Broderick
Keyword(s):  
Three Dimensional ◽  
Equation Model ◽  
Decay Rates ◽  
Phase Portraits ◽  
Stable Phase ◽  
Bifurcation Structure ◽  
Decay Times ◽  
Comprehensive Picture ◽  
Two Parameter ◽  
Parameter Bifurcation

We consider self-pulsing in lasers with a gain section and an absorber section via a mechanism known as [Formula: see text]-switching, as described mathematically by the Yamada ordinary differential equation model for the gain, the absorber and the laser intensity. More specifically, we are interested in the case that gain and absorber decay on different time-scales. We present an overall bifurcation structure by showing how the two-parameter bifurcation diagram in the plane of pump strength versus decay rate of the gain changes with the ratio between the two decay rates. In total, there are ten cases BI to BX of qualitatively different two-parameter bifurcation diagrams, which we present with an explanation of the transitions between them. Moroever, we show for each of the associated eleven cases of structurally stable phase portraits (in open regions of the parameter space) a three-dimensional representation of the organization of phase space by the two-dimensional manifolds of saddle equilibria and saddle periodic orbits. The overall bifurcation structure provides a comprehensive picture of the observable dynamics, including multistability and excitability, which we expect to be of relevance for experimental work on [Formula: see text]-switching lasers with different kinds of saturable absorbers.

scholarly journals Generalizing Source Geometry of Site Contamination by Simulating and Analyzing Analytical Solution of Three-Dimensional Solute Transport Model

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Xingwei Wang ◽  
Jiajun Chen ◽  
Hao Wang ◽  
Jianfei Liu
Keyword(s):  
Solute Transport ◽  
Contaminant Transport ◽  
Transport Model ◽  
Three Dimensional ◽  
Fate And Transport ◽  
Source Area ◽  
Equation Model ◽  
Plume Migration ◽  
Advective Diffusion ◽  
Source Geometry

Due to the uneven distribution of pollutions and blur edge of pollutant area, there will exist uncertainty of source term shape in advective-diffusion equation model of contaminant transport. How to generalize those irregular source terms and deal with those uncertainties is very critical but rarely studied in previous research. In this study, the fate and transport of contaminant from rectangular and elliptic source geometry were simulated based on a three-dimensional analytical solute transport model, and the source geometry generalization guideline was developed by comparing the migration of contaminant. The result indicated that the variation of source area size had no effect on pollution plume migration when the plume migrated as far as five times of source side length. The migration of pollution plume became slower with the increase of aquifer thickness. The contaminant concentration was decreasing with scale factor rising, and the differences among various scale factors became smaller with the distance to field increasing.

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