Dispersion Tensor Research Articles

Hydrodynamic dispersion in Hele-Shaw flows with inhomogeneous wall boundary conditions

Single-phase flow inside a Hele-Shaw cell can exhibit inhomogeneous flow fields, for example when actuated by electroosmosis with varying wall mobilities, leading to internal pressure gradients. We derive a two-dimensional dispersion model for a dissolved species in such a non-uniform flow field, utilizing a multiple-scale perturbation approach. The resulting two-dimensional transport equation is an advection–diffusion equation containing an effective dispersion tensor field and additional advection-correction terms. It can be viewed as a generalization of the well-known Taylor–Aris dispersion model. The dispersion model allows for flow fields with both stationary and oscillatory components. For the special case of non-uniform flow induced by both pressure gradients and electroosmosis, we derive expressions for the flow field in the long-wavelength limit. These include arbitrary, time-dependent functions for both the driving field as well as the wall mobilities. We discuss the general characteristics of the model using a sinusoidally varying wall mobility, and derive analytical expressions for the dispersion tensor. Then, in order to validate the model, we compare three-dimensional Lagrangian particle tracing simulations with the dispersion model for several test cases, including stationary and oscillatory shear flow as well as a recirculating flow field. For each test case, a good agreement between the full three-dimensional simulations and the results of the two-dimensional dispersion model is obtained. The presented model has the potential to significantly simplify computations of mass transport in Hele-Shaw flows.

Macroscopic Model for Passive Mass Dispersion in Porous Media Including Knudsen and Diffusive Slip Effects

In this work, a macroscopic model for incompressible and Newtonian gas flow coupled to Fickian and advective transport of a passive solute in rigid and homogeneous porous media is derived. At the pore-scale, both momentum and mass transport phenomena are coupled, not only by the convective mechanism in the mass transport equation, but also in the solid-fluid interfacial boundary condition. This boundary condition is a generalization of the Kramers-Kistemaker slip condition that includes the Knudsen effects. The resulting upscaled model, applicable in the bulk of the porous medium, corresponds to: 1) A Darcy-type model that involves an apparent permeability tensor, complemented by a dispersive term and 2) A macroscopic convection-dispersion equation for the solute, in which both the macroscopic velocity and the total dispersion tensor are influenced by the slip effects taking place at the pore-scale. The use of the model is restricted by the starting assumptions imposed in the governing equations at the pore scale and by the (spatial and temporal) constraints involved in the upscaling process. The different regimes of application of the model, in terms of the Péclet number values, are discussed as well as its extents and limitations. This new model generalizes previous attempts that only include either Knudsen or diffusive slip effects in porous media.

General dispersion properties of magnetized plasmas with drifting bi-Kappa distributions. DIS-K: Dispersion Solver for Kappa Plasmas

Space plasmas are known to be out of (local) thermodynamic equilibrium, as observations show direct or indirect evidences of non-thermal velocity distributions of plasma particles. Prominent are the anisotropies relative to the magnetic field, anisotropic temperatures, field-aligned beams or drifting populations, but also, the suprathermal populations enhancing the high-energy tails of the observed distributions. Drifting bi-Kappa distribution functions can provide a good representation of these features and enable for a kinetic fundamental description of the dispersion and stability of these collision-poor plasmas, where particle–particle collisions are rare but wave–particle interactions appear to play a dominant role in the dynamics. In the present paper we derive the full set of components of the dispersion tensor for magnetized plasma populations modelled by drifting bi-Kappa distributions. A new solver called DIS-K (DIspersion Solver for Kappa plasmas) is proposed to solve numerically the dispersion relations of high complexity. The solver is validated by comparing with the damped and unstable wave solutions obtained with other codes, operating in the limits of drifting Maxwellian and non-drifting Kappa models. These new theoretical tools enable more realistic characterizations, both analytical and numerical, of wave fluctuations and instabilities in complex kinetic configurations measured in-situ in space plasmas.

Modeling Transport and Adsorption of Arsenic Ions in Iron-Oxide Laden Porous Media. Part I: Theoretical Developments

The process of transport and trapping of arsenic ions in porous water filters is treated as a classic mass transport problem which, at the pore scale, is modeled using the traditional convection-diffusion equation, representing the migration of species present in very small (tracer) amounts in water. The upscaling, conducted using the volume averaging method, reveals the presence of two possible forms of the macroscopic equations for predicting arsenic concentrations in the filters. One is the classic convection-dispersion equation with the total dispersion tensor as its main transport coefficient, and which is obtained from a closure formulation similar to that of the passive diffusion problem. The other equation form includes an additional transport coefficient, hitherto ignored in the literature and identified here as the adsorption-induced vector. These two coefficients in the latter form are determined from a system of two closure problems that include the effects of both the passive diffusion as well as the adsorption of arsenic by the solid phase of the filter. This theoretical effort represents the first serious effort to introduce a detailed micro–macro coupling while modeling the transport of arsenic species in water filters representing homogeneous porous media.

Calibrating the BHB star distance scale and the halo kinematic distance to the Galactic Centre

ABSTRACT We report the first determination of the distance to the Galactic Centre based on the kinematics of halo objects. We apply the statistical-parallax technique to the sample of ∼2500 blue horizontal branch (BHB) stars compiled by Xue et al. to simultaneously constrain the correction factor to the photometric distances of BHB stars as reported by those authors and the distance to the Galactic Centre to find R = 8.2 ± 0.6 kpc. We also find that the average velocity of our BHB star sample in the direction of Galactic rotation, V0 = −240 ± 4 km s−1, is greater by about 20 km s−1 in absolute value than the corresponding velocity for halo RR Lyrae type stars (V0 = −222 ± 4 km s−1) in the Galactocentric distance interval from 6 to 18 kpc, whereas the total (σV) and radial (σr) velocity dispersion of the BHB sample are smaller by about 40–45 km s−1 than the corresponding parameters of the velocity dispersion ellipsoid of halo RR Lyrae type variables. The velocity dispersion tensor of halo BHB stars proved to be markedly less anisotropic than the corresponding tensor for RR Lyrae type variables: the corresponding anisotropy parameter values are equal to βBHB = 0.51 ± 0.02 and βRR = 0.71 ± 0.03, respectively.

Visualization of the Anisotropy of the Velocity Dispersion and Characteristics of the Multi-Velocity Continuum in the Regions of Multi-Stream Flows of Gas-Dust Media with Polydisperse Dust.

Collisionless media devoid of intrinsic stresses, for example, a dispersed phase in a multiphase medium, have a much wider variety of space-time structures and features formed in them than collisional media, for example, a carrier, gas, or liquid phase. This is a consequence of the fact that evolution in such media occurs in phase space, i.e., in a space of greater dimensions than the usual coordinate space. As a consequence, the process of the formation of features in collisionless media (clustering or vice versa, a loss of continuity) can occur primarily in the velocity space, which, in contrast to the features in the coordinate space (folds, caustics, or voids), is poorly observed directly. To identify such features, it is necessary to use visualization methods that allow us to consider, in detail, the evolution of the medium in the velocity space. This article is devoted to the development of techniques that allow visualizing the degree of anisotropy of the velocity fields of collisionless interpenetrating media. Simultaneously tracking the behavior of different fractions in such media is important, as their behavior can be significantly different. We propose three different techniques for visualizing the anisotropy of velocity fields using the example of two- and three-continuum dispersed media models. We proposed the construction of spatial distributions of eccentricity fields (scalar fields), or fields of principal directions of the velocity dispersion tensor (tensor fields). In the first case, we used some simple eccentricity functions for dispersion tensors for two fractions simultaneously, which we call surrogate entropy. In the second case, to visualize the anisotropy of the velocity fields of three fractions simultaneously, we used an ordered array (3-vector) of eccentricities for the color representation through decomposition in three basic colors. In the case of a multi-stream flow, we used cluster analysis methods to identify sections of a multi-stream flow (beams) and used glyphs to visualize the entire set of beams (vector-tensor fields).

Experimental and numerical investigations of spatially-varying dispersion tensors based on vertical velocity profile and depth-averaged flow field

The dispersion coefficient is the most crucial parameter that governs the model prediction of pollutant mixing in rivers. However, the spatial variability of dispersion coefficient associated with velocity structures has not yet been thoroughly investigated. In this study, two types of spatially varying dispersion tensors were considered using both the vertical velocity profiles (DF) and the depth-averaged flow fields (DA), and the pollutant transports using DF and DA were compared with tracer test results in a meandering channel. The results showed that the peak concentration and arrival time were more accurately predicted by using DF than DA, because the flow properties associated with the shear induced dispersion were explicitly incorporated in the DF. The spatial distributions of the DF showed variations in both stream-wise and span-wise directions due to the creation and destruction of secondary flow in the meandering channel. In contrast, the DA only changed in the stream-wise direction. Therefore, the DF is advantageous to reproduce shear dispersion processes.

First Gaia dynamical model of the Milky Way disc with six phase space coordinates: a test for galaxy dynamics

ABSTRACT We construct the first comprehensive dynamical model for the high-quality subset of stellar kinematics of the Milky Way disc, with full 6D phase-space coordinates, provided by the Gaia Data Release 2. We adopt an axisymmetric approximation and use an updated Jeans Anisotropic Modelling (JAM) method, which allows for a generic shape and radial orientation of the velocity ellipsoid, as indicated by the Gaia data, to fit the mean velocities and all three components of the intrinsic velocity dispersion tensor. The Milky Way is the first galaxy for which all intrinsic phase space coordinates are available, and the kinematics are superior to the best integral-field kinematics of external galaxies. This situation removes the long-standing dynamical degeneracies and makes this the first dynamical model highly overconstrained by the kinematics. For these reasons, our ability to fit the data provides a fundamental test for both galaxy dynamics and the mass distribution in the Milky Way disc. We tightly constrain the volume average total density logarithmic slope, in the radial range 3.6–12 kpc, to be αtot = −2.149 ± 0.055 and find that the dark halo slope must be significantly steeper than αDM = −1 (NFW). The dark halo shape is close to spherical and its density is ρDM(R⊙) = 0.0115 ± 0.0020 M⊙ pc−3 (0.437 ± 0.076 GeV cm−3), in agreement with previous estimates. The circular velocity at the solar position vcirc(R⊙) = 236.5 ± 3.1 km s−1 (including systematics) and its gently declining radial trends are also consistent with recent determinations.

Effective equation to assess solute transport in two-lithology reservoirs

Understanding solute transport in heterogeneous porous media is a challenging problem due to the degree of complexities coming from the spatial variations of petrophysical properties. The developments of theoretical formulations are powerful tools used to obtain insights regarding the phenomenology involved in the transport process when analyzing large-scale systems. In this context and within the framework of the volume averaging method, in this work we present an effective one-equation model and its validity conditions for the solute transport in heterogeneous reservoirs characterized by two-lithology arrangements. For this purpose, the hypothesis of local mass equilibrium was assumed valid and what follows included the determination of physical constraints supporting such a hypothesis. In a previous work (Aguilar-Madera et al., 2019), the closure problems to calculate the effective coefficients of the two-equation model were presented, and with this base we present here the numerical estimation of the effective velocity vector and the mass dispersion tensor included in the one-equation model. As one application, the numerical solution of the one-equation model was carried out for different lithology arrangements and the breakthrough curves for two permeability ratio scenarios were analyzed. The results confirm that the permeability ratio dramatically impact not only the tracer dispersion but also the arrival time at observation points, which is strongly correlated to the velocity vector field and its spatial deviations. We found that the local mass equilibrium is met more easily in lenticular reservoirs provided that the boundary and initial conditions have no long-term effects. Finally, the dispersion coefficient was compared against reported reservoir field data, finding similarities in the orders of magnitude for certain geologies.

Entropy: (1) The former trouble with particle-tracking simulation, and (2) A measure of computational information penalty

Traditional random-walk particle-tracking (PT) models of advection and dispersion do not track entropy, because particle masses remain constant. However, newer mass-transfer particle tracking (MTPT) models have the ability to do so because masses of all compounds may change along trajectories. Additionally, the probability mass functions (PMF) of these MTPT models may be compared to continuous solutions with probability density functions, when a consistent definition of entropy (or similarly, the dilution index) is constructed. This definition reveals that every discretized numerical model incurs a computational entropy. Similar to Akaike’s (1974, 1992) entropic penalty for larger numbers of adjustable parameters, the computational complexity of a model (e.g., number of nodes or particles) adds to the entropy and, as such, must be penalized. Application of a new computational information criterion reveals that increased accuracy is not always justified relative to increased computational complexity. The MTPT method can use a particle-collision based kernel or an adaptive kernel derived from smoothed-particle hydrodynamics (SPH). The latter is more representative of a locally well-mixed system (i.e., one in which the dispersion tensor equally represents mixing and solute spreading), while the former better represents the separate processes of mixing versus spreading. We use computational means to demonstrate the fitness of each of these methods for simulating 1-D advective-dispersive transport with uniform coefficients.

Virial models and anisotropy of velocity dispersion in E-galaxies

A tensor virial approach is used to construct detailed models of 22 flattened ellipticals. The models are combined with the new observational data: extended surface brightness distributions, the profiles of isophotes flattening and their twisting, the rotation curves and the velocity dispersion profiles. A key feature of these models is a rigorous consideration of the influence of the spatial shape of galaxies (oblate spheroid or triaxial ellipsoid) on the dynamics, as well as the structure of the internal density layers in these systems. For each galaxy the ratio of the rotational energy to the potential energy was found. Comparing this ratio with the observed it is concluded that the majority of these systems cannot have isotropic velocity dispersion tensors. The anisotropy parameter limited to the interval 0.0-0.224. A correlation was found between and flattening of the outer regions for giant E-galaxies. Our results are compared with the results of other researchers. It was found that for small fast rotators our values \b{eta} are in good agreement with those, obtained in the ATLAS project. However, for giant E-galaxies, our models provide better agreement with observations, than axisymmetric JAM models. In addition, our velocity dispersion anisotropy results are in satisfactory agreement with the results of high resolution cosmological simulations in the Illustris project

Upscaling Reactive Transport Under Hydrodynamic Slip Conditions in Homogeneous Porous Media

AbstractReactive transport of a dilute species in a Newtonian fluid saturating a homogeneous porous medium for slip flow is analyzed in this work. Incompressible Newtonian flow with a first‐order (Navier) slip boundary condition is considered together with a first‐order heterogeneous reaction. Kinetic numbers range up to unity while Knudsen numbers, characteristic of slip‐flow, are smaller than approximately 0.1. The pore‐scale problem is upscaled, using the volume averaging method, to obtain a macroscopic transport equation (referred to as the complete upscaled model) operating at the Darcy scale. Using order of magnitude estimates and an expansion in terms of the Kinetic number, simplifications in the expressions of the effective coefficients are explored. The conditions under which the effective convective velocity and the total dispersion tensor are independent of reaction are derived. Under such conditions, a simplified version of the macroscopic transport equation is obtained. This is referred to as the simplified model. Numerical results for a simple structure indicate that the longitudinal dispersion coefficient decreases with decreasing Knudsen number, while the opposite holds for the transverse component of the total dispersion tensor, the effective convective velocity, and the effective reaction rate coefficient. These effects are more evident when the Péclet number increases (i.e., in the convective‐favored regime) and when slip effects are more pronounced. Results predicted by the simplified and the complete versions of the upscaled model are validated with direct numerical simulations of the pore‐scale problem on a two‐dimensional model structure.

Study of the Effect of Thermal Dispersion on Internal Natural Convection in Porous Media Using Fourier Series

Natural convection in a porous enclosure in the presence of thermal dispersion is investigated. The Fourier–Galerkin (FG) spectral element method is adapted to solve the coupled equations of Darcy’s flow and heat transfer with a full velocity-dependent dispersion tensor, employing the stream function formulation. A sound implementation of the FG method is developed to obtain accurate solutions within affordable computational costs. In the spectral space, the stream function is expressed analytically in terms of temperature, and the spectral system is solved using temperature as the primary unknown. The FG method is compared to finite element solutions obtained using an in-house code (TRACES), OpenGeoSys and COMSOL Multiphysics®. These comparisons show the high accuracy of the FG solution which avoids numerical artifacts related to time and spatial discretization. Several examples having different dispersion coefficients and Rayleigh numbers are tested to analyze the solution behavior and to gain physical insight into the thermal dispersion processes. The effect of thermal dispersion coefficients on heat transfer and convective flow in a porous square cavity has not been investigated previously. Here, taking advantage of the developed FG solution, a detailed parameter sensitivity analysis is carried out to address this gap. In the presence of thermal dispersion, the Rayleigh number mainly affects the convective velocity and the heat flux to the domain. At high Rayleigh numbers, the temperature distribution is mainly controlled by the longitudinal dispersion coefficient. Longitudinal dispersion flux is important along the adiabatic walls while transverse dispersion dominates the heat flux toward the isothermal walls. Correlations between the average Nusselt number and dispersion coefficients are derived for three Rayleigh number regimes.

A dispersive homogenization model for composites and its RVE existence

An asymptotic homogenization model considering wave dispersion in composites is investigated. In this approach, the effect of the microstructure through heterogeneity-induced wave dispersion is characterised by an acceleration gradient term scaled by a “dispersion tensor”. This dispersion tensor is computed within a statistically equivalent representative volume element (RVE). One-dimensional and two-dimensional elastic wave propagation problems are studied. It is found that the dispersive multiscale model shows a considerable improvement over the non-dispersive model in capturing the dynamic response of heterogeneous materials. To test the existence of an RVE for a realistic microstructure for unidirectional fiber-reinforced composites, a statistics study is performed to calculate the homogenized properties with increasing microstructure size. It is found that the convergence of the dispersion tensor is sensitive to the spatial distribution pattern. A calibration study on a composite microstructure with realistic spatial distribution shows that convergence is found although only with a relatively large micromodel.

A Vectorized Regularization Method for Multivalued Parameters Identification

The identification of multivalued parameters is formulated as a constraint minimization problem called primal problem. We embed it in a family of perturbed problems and we associate a dual problem with it using the conjugate functions. Basing on the primal-dual relationship, under some qualification conditions on the parameters to be identified we elaborate the well posedeness, convergence and stability of the solution assuming. Numerical simulations are described in the end for the identification of discontinuous dispersion tensor in transport equations.

Evolution of dark matter velocity dispersion

Cosmological perturbation theory for the late Universe dominated by dark matter is extended beyond the perfect pressureless fluid approximation by taking the dark matter velocity dispersion tensor as an additional field into account. A proper tensor decomposition of the latter leads to two additional scalar fields, as well as a vector and a tensor field. Most importantly, the trace of the velocity dispersion tensor can have a spatially homogeneous expectation value. While it decays at early times, we show that a back-reaction effect quadratic in perturbations makes it grow strongly at late times. We compare sterile neutrinos as a candidate for comparatively warm dark matter to weakly interacting massive particles as a rather cold dark matter candidate and show that the late time growth of velocity dispersion is stronger for the latter. Another feature of a non-vanishing velocity dispersion tensor is that it can account for multiple streams within the fluid description and thereby allows to treat times and scales beyond shell-crossing.

Stellar systems following the R1/m luminosity law

The Sérsic or R1/m model has become the de facto standard model to describe the surface brightness profiles of early-type galaxies and the bulges of spiral galaxies. The photometric, intrinsic, and dynamical properties of this model have been investigated, but mainly for fairly large Sérsic indices m. For small values of m, appropriate for low-mass and dwarf ellipticals, a detailed investigation of these properties is still lacking. In this study, we used a combination of numerical and analytical techniques to investigate the Sérsic model over the entire range of Sérsic parameters, focussing on the small m regime, where a number of interesting and surprising properties are found. For all values m < 1, the model is characterised by a finite central luminosity density, and for m < 1/2, even a central depression in the luminosity density profile. This behaviour translates to the dynamical properties: we show that all Sérsic models with m ⩾ 1/2 can be supported by an isotropic velocity dispersion tensor, and that these isotropic models are stable to both radial and non-radial perturbations. The models with m < 1/2, on the other hand, cannot be supported by an isotropic velocity dispersion tensor.

Large-scale velocity dispersion and the cosmic web

Gravitational collapse in cosmological context produces an intricate cosmic web of voids, walls, filaments and nodes. The anisotropic nature of collisionless collapse leads to the emergence of an anisotropic velocity dispersion, or stress, that absorbs most of the kinetic energy after shell-crossing. In this paper, we measure this large-scale velocity dispersion tensor $\sigma^2_{ij}$ in $N$-body simulations using the phase-space interpolation technique. We study the environmental dependence of the amplitude and anisotropy of the velocity dispersion tensor field, and measure its spatial correlation and alignment. The anisotropy of $\sigma^2_{ij}$ naturally encodes the collapse history and thus leads to a parameter-free identification of the four dynamically distinct cosmic web components. We find this purely dynamical classification to be in good agreement with some of the existing classification methods. In particular, we demonstrate that $\sigma^2_{ij}$ is well aligned with the large-scale tidal field. We further investigate the influence of small scale density fluctuations on the large scale velocity dispersion, and find that the measured amplitude and alignments are dominated by the largest perturbations and thus remain largely unaffected. We anticipate that these results will give important new insight into the anisotropic nature of gravitational collapse on large scales, and the emergence of anisotropic stress in the cosmic web.

Advection–Diffusion in Porous Media with Low Scale Separation: Modelling via Higher-Order Asymptotic Homogenisation

Asymptotic multiple scale homogenisation allows to determine the effective behaviour of a porous medium by starting from the pore-scale description, when there is a large separation between the pore-scale and the macroscopic scale. When the scale ratio is “small but not too small”, the standard approach based on first-order homogenisation may break down since additional terms need to be taken into account in order to obtain an accurate picture of the overall response of the medium. The effect of low scale separation can be obtained by exploiting higher-order equations in the asymptotic homogenisation procedure. The aim of the present study is to investigate higher-order terms up to the third order of the advective–diffusive model to describe advection–diffusion in a macroscopically homogeneous porous medium at low scale separation. The main result of the study is that the low separation of scales induces dispersion effects. In particular, the second-order model is similar to the most currently used phenomenological model of dispersion: it is characterised by a dispersion tensor which can be decomposed into a purely diffusive component and a mechanical dispersion part, whilst this property is not verified in the homogenised dispersion model (obtained at higher Peclet number). The third-order description contains second and third concentration gradient terms, with a fourth-order tensor of diffusion and with a third-order and an additional second-order tensors of dispersion. The analysis of the macroscopic fluxes shows that the second- and the third-order macroscopic fluxes are distinct from the volume averages of the corresponding local fluxes and allows to determine expressions of the non-local effects.

PAR[formula omitted]: Parallel Random Walk Particle Tracking Method for solute transport in porous media

Computational modeling of solute migration in groundwater systems is a fundamental component in water resources management and risk analysis. Therefore, it is imperative to have fast and reliable computational tools to simulate solute transport in groundwater systems. In this work we present PAR2, a GPU-accelerated solute transport simulator based on the Random Walk Particle Tracking (RWPT) technique, a Lagrangian method particularly suited for parallelization. PAR2 is able to run on any computing platform equipped with an NVIDIA GPU, such as common desktops and High-Performance Computing (HPC) nodes. The program is developed in C++/CUDA. In our illustration, groundwater flow is simulated on a structured grid using MODFLOW, which can be linked to PAR2 using the LMT package. Simulation parameters can be defined through a convenient YAML configuration file. Additionally, we propose an analytical treatment of the dispersion tensor that allows the RWPT to be effectively implemented using GPU parallelization. The speedup gained with the parallelization drastically reduces the total simulation time, allowing the application of computationally expensive algorithms (e.g., Monte-Carlo simulation) on large-scale stochastic hydro-systems. Program summaryProgram Title: PAR2.Program Files doi:http://dx.doi.org/10.17632/4pkhgx8wcb.1Licensing provisions: GPLv3Programming language: C++/CUDANature of problem: GPU-accelerated simulation of solute transport in saturated porous media.Solution method: Implementation of the particle tracking method, a fully Lagrangian approach to solve the advection–dispersion equations.