Uniform Initial Conditions Research Articles

Comment on “Series solution for Richards Equation under concentration boundary conditions and uniform initial conditions” by Guido Daniel Salvucci

Reference ECOL-ARTICLE-1998-002doi:10.1029/98WR01834 URL: http://www.agu.org/journals/wr/ Record created on 2005-12-09, modified on 2016-08-08

Problem of the theory of thermal bridges in cryogenic systems

Here is examined the problem of nonstationary heat conduction for a uniform initial temperature distribution and type-III boundary conditions, which is currently critical to cryogenic systems

Long-time behavior and coexistence in a mutually catalytic branching model

We study a system of two interacting populations which undergo random migration and mutually catalytic branching. The branching rate of one population at a site is proportional to the mass of the other population at the site. The system is modelled by an infinite system of stochastic differential equations, allowing symmetric Markov migration, if the set of sites is discrete $(\mathbb{Z}^d)$, or by a stochastic partial differential equation with Brownian migration if the set of sites is the real line. A duality technique of Leonid Mytnik, which gives uniqueness in law, is used to examine the long-time behavior of the solutions. For example, with uniform initial conditions, the process converges to an equilibrium distribution as $t \to \infty$, and there is coexistence of types in the equilibrium “iff ” the random migration is transient.

Stable computation of turbulent flows with a low-Reynolds-number k-ϵ turbulence model and explicit solver

It is well known that the incorporation of two-equation turbulence models into explicit time-marching schemes, especially the low Reynolds versions, encounters various numerical difficulties. This paper sets out to review the numerical techniques presently used and to analyse the problems encountered. A numerical approach is proposed for the implementation of the low-Reynolds-number (LRN) k—ϵ models into an explicit Runge—Kutta time-marching solver. In this method, a semi-implicit treatment of the source terms of the k and ϵ equations is introduced, and modified residuals of the two equations are used in order to adopt implicit residual smoothing. These techniques ensure the positivity of k and ϵ in the computation, even with uniform initial conditions and local time stepping. The asymptotic behaviour of the turbulence quantities during time marching is analysed, providing insight into the stability and convergence characteristics of the k and ϵ equations using explicit and implicit schemes with and without the damping terms. It is found that the time step size has a strong influence on the evolution of the turbulence quantities and the stability of the solution, whether implicit or explicit schemes are used. The time step size determined by the stability analysis is a function of k, ϵ and their damping terms. This analysis also provides a guide for the selection of time step size for the turbulence equations. It is found that the selection of time step size is an important factor in ensuring realistic values of k, ϵ and μ t in the flow field during time marching from uniform initial conditions. Numerical solutions of several flow test cases are given and the computed results are compared with experimental measurements or theoretical solutions for validation of the method. It is shown that this method is reasonably efficient and robust. © 1997 Elsevier Science Limited.

New approximate analytical technique to solve Richards Equation for arbitrary surface boundary conditions

A general approximation for the solution to the one‐dimensional Richards equation is presented. It applies to arbitrary soil properties and boundary conditions but only uniform initial conditions at current stage. The result is very accurate (within 2%) when the diffusivity is constant, suggesting that the present general formulation is reliable, since the approximation becomes increasingly accurate as the soil‐water diffusivity approaches a delta function.

Series Solution for Richards Equation Under Concentration Boundary Conditions and Uniform Initial Conditions

A series solution which expresses the moisture profiles and surface fluxes during infiltration and exfiltration is derived. The solution applies for conditions of fixed surface moisture content and uniform initial moisture content within an unbounded, homogeneous, nonhysteretic soil column. The series derivation is similar to J. R. Philip's but with a transformed time variable. The series converges for infiltration problems involving soils which exhibit steep increase in capillary pressure near saturation. Under these conditions the solution captures the transition from early square root of time behavior to the longtime traveling wave solution without the use of approximate joining techniques. The soil water retention models for which convergence occurs include the widely applied Brooks‐Corey and van Genuchten models, with minor restrictions on the shape parameters of the latter. Other cases examined include infiltration and exfiltration assuming power law, linear, quadratic, and constant dependence of diffusivity and conductivity on moisture content. For these cases the convergence limits were finite. The solution is particularly well suited for infiltration into sandy soils, for which the transition between short‐time and longtime behavior occurs relatively early.

A Note on Horizontal Redistribution with Capillary Hysteresis

In a recent paper, Philip analyzed horizontal redistribution with capillary hysteresis. He considered the special case for which one semi‐infinite half of a column has an initial condition corresponding to the point where the boundary wetting curve joins the boundary drying curve and the other semi‐infinite half of the column has an initial condition corresponding to the point where the boundary drying curve joins the boundary wetting curve. The similarity solution of this problem developed by Philip can be generalized to cases involving any uniform initial conditions of the semi‐infinite halves of the column, corresponding to arbitrary histories of the pressure head and the water content. For any such pair of initial conditions for the semi‐infinite halves, one of the following three possibilities will apply: (1) no flow between the two halves; (2) conventional flow from the wet half to the dry half; or (3) nonconventional flow from the dry half to the wet half.

The formation of close binary systems

A viable solution to the origin of close binary systems, unaccounted for in recent theories, is presented. Fragmentation, occurring at the end of the secondary collapse phase (during which molecular hydrogen is dissociating), can form binary systems with separations less than 1 au. Two fragmentation modes are found to occur after the collapse is halted. The first consists of the fragmentation of a protostellar disc due to rotational instabilities in a protostellar core, involving both an $m=1$ and an $m=2$ mode. This fragmentation mechanism is found to be insensitive to the initial density distribution: it can occur in both centrally condensed and uniform initial conditions. The second fragmentation mode involves the formation of a rapidly rotating core at the end of the collapse phase which is unstable to the axisymmetric perturbations. This core bounces into a ring which quickly fragments into several components. The binary systems thus formed contain less than 1 per cent of a solar mass and therefore will need to accrete most of their final mass if they are to form a binary star system. Their orbital properties will thus be determined by the properties of the accreted matter.

Statistical moments of reactive solute concentration in a heterogeneous aquifer

Neglecting local solute dispersion, we prove that under realistic conditions the ensemble moments of the local concentration satisfy a mean convection dispersion equation (CDE) for both conservative and instantaneously adsorbing solutes, provided that the solute velocity field is stochastically stationary. Thus the same algorithm can be used to model the ensemble mean concentration and its uncertainty. For uniform initial conditions the solution of the mean CDE contains all the information about the local concentration probability distribution function, which is not Gaussian. The governing equation for k‐point ensemble moments of the concentration is similar in form to the mean CDE, with the same effective dispersion coefficient and convective velocity. The special case of an impulse initial condition is used to illustrate that volume averaging can significantly change the concentration coefficient of variation except at the plume front. The persistence in the longitudinal direction of concentration moments and concentration k‐point moments is caused by large longitudinal megadispersion. Finally, it is suggested that “ergodicity” should be interpreted operationally in terms of an acceptably small coefficient of variation of the concentration field so that this concept can be applied to experimental field data.

CONNECTIVITY AND THE DYNAMICS OF INTEGRATE-AND-FIRE NEURAL NETWORKS

The dynamics following periodic stimulation at a single point with uniform initial conditions, of an excitatory network of identical integrate-and-fire neurons is investigated as a function of interconnectivity. It is shown that propagating spiral waves as well as other forms of self-maintained activity can arise provided that the probability of neural interconnectivity is an exponentially decreasing function of interneuronal distance. Other forms of connectivity, e.g. nearest neighbor, step function, do not produce spirals.

Flux‐Averaged Concentrations for Transport in Soils Having Nonuniform Initial Solute Distributions

AbstractThe need to distinguish between volume‐averaged or resident concentrations (cr) and flux‐averaged or flowing concentrations (cf) is now widely accepted. Flux‐averaged concentrations associated with the convection‐dispersion equation (CDE) have been mostly used for solute transport problems involving uniform initial distributions. We present flux‐averaged concentrations for nonuniform initial distributions using analytical solution methods for a semi‐infinite soil system and numerical methods for a finite system. Mathematically, cf is equivalent to cr associated with a first‐type inlet condition (rather than a third‐type condition) only for semi‐infinite soil profiles having uniform initial conditions. We show that, for a stepwise initial distribution, cf can be both negative or much greater than the initial concentration of cr, especially during the early stages of solute displacement. This physically odd situation results from the fact that cf represents a solute flux rather than a directly measurable volumetric concentration. Flux‐averaged concentrations at the exit of a finite soil column with a uniform initial distribution are nearly identical to cf for a semi‐infinite system when the column Peclet number is greater than ≈ 5. However, if the initial distribution involves a high gradient in cr near the exit, cr values for finite and semi‐infinite systems at the exit can be very different, similarly as those for cr, because of the adoption of different outlet conditions.

A similarity solution for two‐phase water, air, and heat flow near a linear heat source in a porous medium

Placement of a heat source in a partially saturated geologic medium causes strongly coupled thermal and hydrologic behavior. To study this problem, a recently developed semianalytical solution for two‐phase flow of water and heat in a porous medium has been extended to include an air component and to incorporate several physical effects that broaden its range of applicability. The problem considered is the placement of a constant‐strength linear heat source in an infinite homogeneous medium with uniform initial conditions. Under these conditions the governing partial differential equations in radial distance r and time t reduce to ordinary differential equations through the introduction of a similarity variable η = r/t1/2. The resulting equations are coupled and nonlinear, necessitating a numerical integration. The similarity solution developed here is used to investigate various physical phenomena related to partially saturated flow in low‐permeability rock, such as vapor pressure lowering, pore level phase change effects, and an effective continuum representation of fractured/porous media. Application to several illustrative problems arising in the context of high‐level nuclear waste disposal at Yucca Mountain, Nevada, indicates that fluid flow, phase changes, and latent heat transfer may have a significant impact on conditions at the repository.

Geometry and dynamics of deterministic sand piles.

We report a study of the relaxation process of Bak-Tang-Wiesenfeld's sand pile under uniform initial conditions. For most geometries and initial conditions the final states consist of intricate geometric patterns, some of which are self-similar. Even without randomness, the flow of sand during relaxation often displays 1/f behavior, and this arises from interactions between the diffusive flow and the development of the final pattern.

Hindered settling and hydrodynamic dispersion in quiescent sedimenting suspensions

The motion of an individual sphere settling in the midst of a suspension of like spheres has been examined experimentally for suspensions with volume concentrations, φ, of 2.5–10%, under creeping flow conditions and in the absence of Brownian motion. In the experiments, silvered glass spheres were tracked in optically transparent suspensions of glass beads. Arrival times measured at a series of horizontal planes were converted into average settling speeds. These average speeds yield the hindered settling speed as a function of concentration. The hindered settling speed, normalized by the isolated sphere settling speed, exhibits a 1 − 4φ + 8φ 2 dependence for the range of concentrations investigated. The settling speed fluctuations are quite large, ranging up to 46% of the average, and have long-time (large settling distance) behavior characteristic of a Fickian diffusion process. Dispersion coefficients have therefore been determined from the asymptotic dependence upon settling distance of the variance in settling speed. These coefficients scale with the product of the hindered settling speed and the sphere radius. The dimensionless dispersion coefficients, all O(1), increase with concentration for φ < 5%, then slightly decrease at higher concentrations. Verification of the scaling through the use of two particle sizes, care taken to mix the suspensions to random, uniform initial conditions, and the robustness of the statistics over many realizations preclude the possibility of this phenomenon being an experimental artifact and support the hypothesis that hydrodynamic dispersion of suspended particles will result from viscous interactions between the particles.

A semianalytical solution for heat-pipe effects near high-level nuclear waste packages buried in partially saturated geological media

The emplacement of a strong heat source, such as a high-level nuclear waste package, in a partially saturated permeable medium will give rise to the development of a heat pipe. The present paper analyzes a simplified version of this problem that has a steady state solution, for radial geometry. The solution is obtained in semianalytical form, and is compared to the analogous solution for a linear heat pipe. Various applications are presented for porous as well as for fractured-porous media with different hydrologie properties. The parameters determining heat-pipe length and the question of whether the vicinity of the heat source dries up are discussed. The semianalytical solution is verified by numerical simulations that show the transient evolution from uniform initial conditions to the eventual steady state.

Protostellar formation in rotating interstellar clouds. VI - Nonuniform initial conditions

The collapse and fragmentation of rotating protostellar clouds is explored, starting from nonuniform density and nonuniform rotation initial conditions. Whether binary fragmentation occurs during the first dynamic collapse phase depends strongly on the initial density profile. Exponential clouds are only somewhat more resistant to fragmentation than uniform-density clouds, but power-law clouds do not undergo fragmentation for likely values of a relevant parameter. Because binary fragments start from profiles intermediate between uniform density and exponential clouds, minimum protostellar mass for population I stars should be increased to approximately 0.02 solar mass. The axisymmetric Terey et al. (1984) model should be stable with respect to nonaxisymmetric perturbations. Considering the observed binary frequency, collapse from power-law initial conditions appears to be less common than collapse from more uniform initial conditions.

A Simple Analytical Solution for the Boussinesq One‐Dimensional Groundwater Flow Equation

An approximate analytical solution to the Boussinesq equation is presented here. Uniform initial conditions and a step function increase of piezometric head on the boundary are assumed. The one‐dimensional problem is reduced to an ordinary differential equation through Boltzmann's transformation, and a technique exploiting some basic characteristics of the exact solution leads to an approximate polynomial solution of the problem. The technique presented here can also be applied to one‐dimensional nonlinear diffusion problems. A numerical procedure incorporating some of the analytical characteristics of the exact solution is presented. A discussion concludes the paper.

Self-similar temperature, pressure and moisture distributions within an intensely heated porous half space

A model for the transient heat and mass transfer within an intensely heated porous half space is presented. The model is based on three principal transport phenomena: heat conduction, vapor convection under pressure gradients and the evaporation-recondensation mechanism. Local liquid-vapor equilibrium is assumed. The governing equations consist of two sets: one for the porous region that has dried and one for the remaining moist region. A self-similar solution is obtained for the combination of uniform initial conditions with a step change in boundary conditions. Transient temperature, pressure and moisture distributions are described in closed form for the dry region and numerically for the moist region. The model was applied to study the rate of moisture loss from a drying concrete wall. The results correspond well with results of previously reported investigations.

Collapse, Equilibrium, and Fragmentation of Rotating, Adiabatic Clouds

AbstractNumerical calculations of the collapse of adiabatic clouds from uniform density and rotation initial conditions show that when restricted to axisymmetry, the clouds form either near-equilibrium spheroids or rings. Rings form in the collapse of low thermal energy clouds and have β = T/|w|≳ 0.43. When the axisymmetric constraint is removed and an initial m=2 density variation is introduced, clouds either collapse to form near-equilibrium ellipsoids or else fragment into binary systems through a bar phase. Ellipsoids form in the collapse of high thermal energy clouds and have β ≲ 0.27. The results are consistent with the critical values of β for instabilities in Maclaurin spheroids, and suggest that protostellar clouds may undergo a dynamic fragmentation in the nonisothermal collapse regime.

Collapse, equilibrium, and fragmentation of rotating, adiabatic clouds

Numerical calculations of the collapse of adiabatic clouds from uniform density and rotation initial conditions show that when restricted to axisymmetry, the clouds form either near-equilibrium spheroids or rings. Rings form in the collapse of low thermal energy clouds and have β = T/ ¦W¦≳ 0.43. When the axisymmetric constraint is removed and an initial m=2 density variation is introduced, clouds either collapse to form near-equilibrium ellipsoids or else fragment into binary systems through a bar phase. Ellipsoids form in the collapse of high thermal energy clouds and have 3 ≲ 0.27. The results are consistent with the critical values of β for instabilities in Maclaurin spheroids, and suggest that protostellar clouds may undergo a dynamic fragmentation in the nonisothermal collapse regime.