Multidimensional Water Flow in Variably Saturated Soils

2003 ◽  
Author(s):  
Arthur W. Warrick
Keyword(s):  
Point Source ◽  
Three Dimensional ◽  
Practical Interest ◽  
Two Dimensions ◽  
Point Sources ◽  
Radial Coordinate ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Divergent Flow ◽  
Suction Sampler

Chapters 4 and 5 dealt with one-dimensional rectilinear flow, with and without the effect of gravity. Now the focus is on multidimensional flow. We will refer to two- and three-dimensional flow based on the number of Cartesian coordinates necessary to describe the problem. For this convention, a point source emitting a volume of water per unit time results in a three-dimensional problem even if it can be described with a single spherical coordinate. Similarly, a line source would be two-dimensional even if it could be described with a single radial coordinate. A problem with axial symmetry will be termed a three-dimensional problem even when only a depth and radius are needed to describe the geometry. The pressure at a point source is undefined. But more generally, three-dimensional point sources refer to flow from finite-sized sources into a larger soil domain, such as infiltration from a small surface pond into the soil. Often, the soil domain can be taken as infinite in one or more directions. Also, a point sink can occur with flow to a sump or to a suction sampler. In two dimensions, the same types of example can be given, but we will refer to them as line sources or sinks. Practical interest in point sources includes analyses of surface or subsurface leaks and of trickle (drip) irrigation. The desirability of determining soil properties in situ has provided the impetus for a rigorous analysis of disctension and borehole infiltrometers. Also, environmental monitoring with suction cups or candles, pan lysimeters, and wicking devices all include convergent or divergent flow in multidimensions. There are some conceptual differences between line and point sources and one-dimensional sources. For discussion, consider water supplied at a constant matric potential into drier surroundings. For a one-dimensional source, the corresponding physical problem includes a planar source over an area large enough for “edge” effects to be negligible. For two dimensions, the source might be a long horizontal cylinder or a furrow of finite depth from which water flows. For three dimensions, the source could be a small orifice providing water at a finite rate or a small, shallow pond on the soil surface.

Diffraction of Electrons by Two and Three Mutually Orthogonal Standing Light Waves

1975 ◽  
Vol 53 (2) ◽  
pp. 157-164 ◽  
Author(s):  
F. Ehlotzky
Keyword(s):  
Electron Scattering ◽  
Light Wave ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Standing Light Wave ◽  
Light Waves ◽  
The One ◽  
Rectangular Lattices

The one-dimensional problem of electron scattering by a standing light wave, known as the Kapitza–Dirac effect, is shown to be easily extendable to two and three dimensions, thus showing all characteristics of diffraction of electrons by simple two- and three-dimensional rectangular lattices.

scholarly journals Wave-activity conservation laws for the three-dimensional anelastic and Boussinesq equations with a horizontally homogeneous background flow

2007 ◽  
Vol 594 ◽  
pp. 493-506 ◽  
Author(s):  
TIFFANY A. SHAW ◽  
THEODORE G. SHEPHERD
Keyword(s):  
Conservation Laws ◽  
Large Scale ◽  
Three Dimensional ◽  
Phase Speed ◽  
Wave Activity ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Dimensional Problem ◽  
Background Flow ◽  
Homogeneous Background

Wave-activity conservation laws are key to understanding wave propagation in inhomogeneous environments. Their most general formulation follows from the Hamiltonian structure of geophysical fluid dynamics. For large-scale atmospheric dynamics, the Eliassen–Palm wave activity is a well-known example and is central to theoretical analysis. On the mesoscale, while such conservation laws have been worked out in two dimensions, their application to a horizontally homogeneous background flow in three dimensions fails because of a degeneracy created by the absence of a background potential vorticity gradient. Earlier three-dimensional results based on linear WKB theory considered only Doppler-shifted gravity waves, not waves in a stratified shear flow. Consideration of a background flow depending only on altitude is motivated by the parameterization of subgrid-scales in climate models where there is an imposed separation of horizontal length and time scales, but vertical coupling within each column. Here we show how this degeneracy can be overcome and wave-activity conservation laws derived for three-dimensional disturbances to a horizontally homogeneous background flow. Explicit expressions for pseudoenergy and pseudomomentum in the anelastic and Boussinesq models are derived, and it is shown how the previously derived relations for the two-dimensional problem can be treated as a limiting case of the three-dimensional problem. The results also generalize earlier three-dimensional results in that there is no slowly varying WKB-type requirement on the background flow, and the results are extendable to finite amplitude. The relationship $A^{\cal E}\,{=}\,cA^{\cal P}$ between pseudoenergy $A^{\cal E}$ and pseudomomentum $A^{\cal P}$, where c is the horizontal phase speed in the direction of symmetry associated with $A^{\cal P}$, has important applications to gravity-wave parameterization and provides a generalized statement of the first Eliassen–Palm theorem.

A Comparison of Diffusion Flame Stability in One and Two Spatial Dimensions Near Cold, Inert Surfaces

2001 ◽  
Author(s):  
Robert Vance ◽  
Indrek S. Wichman
Keyword(s):  
Two Dimensions ◽  
Low Flow ◽  
Dimensional System ◽  
Flame Stability ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Cellular Flames ◽  
The One ◽  
Inert Surfaces

Abstract A linear stability analysis is performed on two simplified models representing a one-dimensional flame between oxidizer and fuel reservoirs and a two-dimensional “edge-flame” between the same reservoirs but above a cold, inert wall. Comparison of the eigenvalue spectra for both models is performed to discern the validity of extending the results from the one-dimensional problem to the two-dimensional problem. Of primary interest is the influence on flame stability of thermal-diffusive imbalances, i.e. non-unity Lewis numbers. Flame oscillations are observed when Le > 1, and cellular flames are witnessed when Le < 1. It is found that when Le > 1 the characteristics of flame behavior are consistent between the two models. Furthermore, when Le < 1, the models are found to be in good agreement with respect to the magnitude of the critical wave numbers. Results from the coarse mesh analysis of the two-dimensional system are presented and compared to the one-dimensional eigenvalue spectra. Additionally, an examination of low reactant convection is undertaken. It is concluded that for low flow rates the behavior in one and two dimensions are similar qualitatively and quantitatively.

On the Use of Point Source Solutions for Forced Air Cooling of Electronic Components—Part I: Thermal Wake Models for Rectangular Heat Sources

2003 ◽  
Vol 125 (2) ◽  
pp. 226-234 ◽  
Author(s):  
Alfonso Ortega ◽  
Shankar Ramanathan
Keyword(s):  
Temperature Field ◽  
Point Source ◽  
Three Dimensional ◽  
Air Cooling ◽  
Heat Sources ◽  
Multiple Sources ◽  
One Dimensional ◽  
Forced Air ◽  
The Individual ◽  
Thermal Wake

Analytical solutions are presented for the temperature field that arises from the application of a source of heat on an adiabatic plate or board when the fluid is represented as a uniform flow with an effective turbulent diffusivity, i.e., the so-called UFED flow model. Solutions are summarized for a point source, a one-dimensional strip source, and a rectangular source of heat. The ability to superpose the individual kernel solutions to obtain the temperature field due to multiple sources is demonstrated. The point source solution reveals that the N−1 law commonly observed for the centerline thermal wake decay for three-dimensional arrays is predicted by the point source solution for the UFED model. Examination of the solution for rectangular sources shows that the thermal wake approaches the point source behavior downstream from the source, suggesting a new scaling for the far thermal wake based on the total component power and a length scale given by ε/U. The new scaling successfully collapses the thermal wake for several sizes of components and provides a fundamental basis for experimental observations previously made for arrays of three-dimensional components.

A computational model of the collective fluid dynamics of motile micro-organisms

2002 ◽  
Vol 455 ◽  
pp. 149-174 ◽  
Author(s):  
MATTHEW M. HOPKINS ◽  
LISA J. FAUCI
Keyword(s):  
Fluid Dynamics ◽  
Stokes Equations ◽  
Temporal Dynamics ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Point Sources ◽  
Physical Factors ◽  
Modelling Framework ◽  
Long Time ◽  
Micro Organisms

A mathematical model and numerical method for studying the collective dynamics of geotactic, gyrotactic and chemotactic micro-organisms immersed in a viscous fluid is presented. The Navier–Stokes equations of fluid dynamics are solved in the presence of a discrete collection of micro-organisms. These microbes act as point sources of gravitational force in the fluid equations, and thus affect the fluid flow. Physical factors, e.g. vorticity and gravity, as well as sensory factors affect swimming speed and direction. In the case of chemotactic microbes, the swimming orientation is a function of a molecular field. In the model considered here, the molecules are a nutrient whose consumption results in an upward gradient of concentration that drives its downward diffusion. The resultant upward chemotactically induced accumulation of cells results in (Rayleigh–Taylor) instability and eventually in steady or chaotic convection that transports molecules and affects the translocation of organisms. Computational results that examine the long-time behaviour of the full nonlinear system are presented.The actual dynamical system consisting of fluid and suspended swimming organisms is obviously three-dimensional, as are the basic modelling equations. While the computations presented in this paper are two-dimensional, they provide results that match remarkably well the spatial patterns and long-time temporal dynamics of actual experiments; various physically applicable assumptions yield steady states, chaotic states, and bottom-standing plumes. The simplified representation of microbes as point particles allows the variation of input parameters and modelling details, while performing calculations with very large numbers of particles (≈104–105), enough so that realistic cell concentrations and macroscopic fluid effects can be modelled with one particle representing one microbe, rather than some collection of microbes. It is demonstrated that this modelling framework can be used to test hypotheses concerning the coupled effects of microbial behaviour, fluid dynamics and molecular mixing. Thus, not only are insights provided into the differing dynamics concerning purely geotactic and gyrotactic microbes, the dynamics of competing strategies for chemotaxis, but it is demonstrated that relatively economical explorations in two dimensions can deliver striking insights and distinguish among hypotheses.

Inverse scattering for geophysical problems. III. On the velocity-in version problems of acoustics

1985 ◽  
Vol 399 (1816) ◽  
pp. 153-166 ◽  
Keyword(s):  
Speed Of Sound ◽  
Three Dimensional ◽  
Acoustic Medium ◽  
Inversion Problem ◽  
Dimensional Problem ◽  
Velocity Inversion ◽  
One Dimensional ◽  
Time Harmonic ◽  
Acoustic Media ◽  
Unknown Region

A bounded inhomogeneity D is immersed in an acoustic medium; the speed of sound is a function of position in D , and is constant outside. A time-harmonic source is placed at a point y and the pressure at a point x is measured. Given such measurements at all for all x ∈ P , for all y ∈ P where P is a plane that does not intersect D , can the speed of sound (in the unknown region D ) be recovered? This is a velocity-inversion problem. The three-dimensional problem has been solved analytically by Ramm ( Phys. Lett . 99A, 258-260 (1983)). In the present paper, analogous one-dimensional and two-dimensional problems are solved, as well as the problem where the plane P is the interface between two different acoustic media.

The transport equation with plane symmetry and isotropic scattering

1964 ◽  
Vol 60 (4) ◽  
pp. 909-921 ◽  
Author(s):  
M. G. Smith
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Point Sources ◽  
The Other ◽  
Isotropic Scattering ◽  
Dimensional Problem ◽  
Plane Symmetry ◽  
Double Integral ◽  
One Dimensional ◽  
The One

AbstractThe double integral equation, which takes the place of the Milne equation in the one-dimensional problem, is derived from the governing partial differentio-integral equations. An analytical solution of the problem of a distribution of point sources on a plane, when the other boundaries are at infinity, is then found. The possibility of more complicated boundary conditions is discussed.

scholarly journals Two-scale mathematics and fractional calculus for thermodynamics

2019 ◽  
Vol 23 (4) ◽  
pp. 2131-2133 ◽  
Author(s):  
Ji-Huan He ◽  
Fei-Yu Ji
Keyword(s):  
Fractional Calculus ◽  
Three Dimensional ◽  
The Other ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Dimensional Approach ◽  
One Dimensional ◽  
Fractal Calculus ◽  
Lower Dimensional ◽  
The Continuum

A three dimensional problem can be approximated by either a two-dimensional or one-dimensional case, but some information will be lost. To reveal the lost information due to the lower dimensional approach, two-scale mathematics is needed. Generally one scale is established by usage where traditional calculus works, and the other scale is for revealing the lost information where the continuum assumption might be forbidden, and fractional calculus or fractal calculus has to be used. The two-scale transform can approximately convert the fractional calculus into its traditional partner, making the two-scale thermodynamics much promising.

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