Dispersion Tensor Research Articles

A volume‐averaging and stochastic turbulence modeling framework for homogeneous roughness‐induced drag in turbulent flows

AbstractThe goal of this work is the formulation of a model for the parameterization of homogeneous roughness‐induced drag in turbulent flow simulations. We characterize rough surfaces using their surface‐area moments of roughness‐peaks. Additionally, we consider a probabilistic self‐similar power law distribution for the roughness heights, in such a way that an assumed set of discrete roughness elements can behave as a probabilistic fractal set. Upon these assumptions, the surface characterization is complete and yields wall‐normal profiles of porosity, average roughness element diameter, and equivalent pore diameter of the roughness affected porous flow. The profiles are supplied to a (pore‐based) Reynolds number and porosity‐based drag parameterization for staggered cylinder arrays available in the literature, as well as a one‐dimensional map‐based turbulence model, the one‐dimensional turbulence (ODT) model. The purpose of the latter is a way to provide the missing effects of the equivalent dispersion tensor in the porous flow close to the rough surface, as well as that of the nonlinear turbulent transport away from the wall. Simulations are then carried out in order to evaluate the effects of two different rough surfaces on a turbulent channel flow, comparing results with existing direct numerical simulation (DNS) data. Overall, reasonable agreement is obtained, which suggests that, by using the proposed model, it is possible to model the effects of rough surfaces on wall‐bounded turbulent flows, without the explicit need of the full topological representation of the surface on the numerical domain. To that extent, one clear flow control application for the suggested approach would be that of drag‐based surface optimization.

Evaluation of a Volume-Averaged Species Transport Model with Micro-Macro Coupling for Breakthrough Curve Prediction.

In porous water filters, the transport and entrapment of contaminants can be modeled as a classic mass transport problem, which employs the conventional convection-dispersion equation to predict the transport of species existing in trace amounts. Using the volume-averaging method (VAM), the upscaling has revealed two possible macroscopic equations for predicting contaminant concentrations in the filters. The first equation is the classical convection-dispersion equation, which incorporates a total dispersion tensor. The second equation involves an additional transport coefficient, identified as the adsorption-induced vector. In this study, the aforementioned equations were solved in 1D for column tests using 3D unit cells. The simulated breakthrough curves (BTCs), using the proposed micro-macro-coupling-based VAM model, are compared with the direct numerical simulation (DNS) results based on BCC-type unit cells arranged one-after-another in a daisy chain manner, as well as with three previously reported experimental works, in which the functionalized zeolite and zero-valent iron fillings were used as an adsorbent to remove phosphorous and arsenic from water, respectively. The disagreement of VAM BTC predictions with DNS and experimental results reveals the need for an alternative closure formulation in VAM. Detailed investigations reveal time constraint violations in all the three cases, suggesting this as the main cause of VAM's failure.

Dispersivity calculation in digital twins of multiscale porous materials using the micro-continuum approach

The micro-continuum method is a novel approach to simulate flow and transport in multiscale porous materials. For such materials, the domain can be divided into three sub-domains depending on the local porosity ε: fully resolved solid phase, for which ε = 0, fully resolved pores, for which ε = 1.0, and unresolved pores, for which 0 < ε < 1.0. For such domains, the flow can be solved using the Darcy-Brinkman-Stokes (DBS) equation, which offers a seamless transition between unresolved pores, where flow is described by Darcy’s law, and resolved pores, where flow is described by the Stokes equations. Species transport can then be modelled using a volume-averaged equation. In this work, we present a derivation of the closure problem for the micro-continuum approach. Effective dispersivity tensors can then be calculated through a multi-stage process. First, high resolution images are chosen for characterizing the structure of the unresolved pores. Porosity, permeability and effective dispersivity for the unresolved parts are calculated by solving a closure problem based on Direct Numerical Simulation (DNS) in the high-resolution images. The effective dispersivity is then expressed as a function of the Péclet number, which describes the ratio of advective to diffusive transport. This relationship, along with porosity and permeability, is then integrated into the multiscale domain and the effective dispersivity tensor for the full image is calculated. Our novel method is validated by comparison with the numerical solution obtained for a fully-resolved simulation in a multiscale 2D micromodel. It is then applied to obtain an effective dispersivity model in digital twins for two multiscale materials: hierarchical ceramic foams and microporous carbonate rocks.

A phase-splitting approach to describe macroscopically non-equilibrium transport in porous media

Classical macro-scale dispersion equation cannot correctly represent non-local, non-equilibrium effects, and alternatives had to be proposed to overcome these discrepancies. Two-equation models have been widely used and proved useful to incorporate, to some extent, local non-equilibrium effects. More accurate descriptions may be obtained through the introduction of additional complex terms in the one-equation dispersion equation, e.g. spatial and time convolutions. An alternative more practical route rely on the introduction of a N-equations description. For instance, instead of a two-temperature model, transfers in the solid and/or fluid phase maybe represented by several equations, each one defined by some splitting framework. Splitting of the solid phase has been proposed in the past (multi-rate models). Splitting of the flowing phase has a rustic illustration in the case of the Coats and Smith model, i.e., ”flowing” and ”stagnant” fluid equations. In this paper, we explore further the possibilities offered by N-equations splitting of the flowing phase. The N-equations macro-scale model is developed applying an averaging technique to N-pseudo phases obtained by splitting the flowing phase based on different criteria, e.g. the velocity histogram. Most important model properties are the matrices of dispersion tensors and exchange coefficients, which are provided by a system of specific ”closure” problems. To illustrate this approach, the methodology is applied to Taylor’s dispersion problem, in which non-local effects are produced by pore-scale spatially distributed inlet conditions. The proposed N-equations description was compared to estimates of the N-temperatures obtained from direct numerical simulations. The N-equations model was able to reproduce accurately the dynamic of pore-scale computations, for such reputedly non-homogenizable problems, with a relatively small number of equations.

Modeling and simulation of pollution transport in the Mediterranean Sea using enriched finite element method

This paper presents a novel numerical method for simulating the transport and dispersion of pollutants in the Mediterranean sea. The governing mathematical equations consist of a barotropic ocean model with friction terms, bathymetric forces, Coriolis and wind stresses coupled to an advection–diffusion equation with anisotropic dispersion tensor and source terms. The proposed numerical solver uses a multilevel adaptive semi-Lagrangian finite element method that combines various techniques, including the modified method of characteristics, finite element discretization, coupled projection scheme based on a rotational pressure correction algorithm, and an adaptive L2-projection. The approach employs the gradient of the concentration as an error indicator for enrichment adaptations and increasing the number of quadrature points where needed without refining the mesh. The method is shown to provide accurate and efficient simulations for pollution transport in the Mediterranean sea. The proposed approach distinguishes itself from the well-established adaptive finite element methods for incompressible viscous flows by retaining the same structure and dimension of linear systems during the adaptation process.

Anisotropy ansatz for the axisymmetric Jeans equations

Abstract The Jeans equations do not form a closed system, and to solve them a parametrization relating the velocity moments is often adopted. For axisymmetric models, a phenomenological choice (the “b-ansatz”) is widely used for the relation between the vertical ($\sigma _z^2$) and radial ($\sigma _R^2$) components of the velocity dispersion tensor, thus breaking their identity present in two-integral systems. However, the way in which the ansatz affects the resulting kinematical fields can be quite complicated, so that the analysis of these fields is usually performed only after numerically computing them. We present here a general procedure to study the properties of the ansatz-dependent fields $\overline{v_{\varphi }^2}$, $\Delta =\overline{v_{\varphi }^2}- \sigma _z^2$ and $\Delta _R= \overline{v_{\varphi }^2}- \sigma _R^2$. Specifically, the effects of the b-ansatz can be determined before solving the Jeans equations once the behaviour over the (R, z)-plane of three easy-to-build ansatz-independent functions is known. The procedure also constrains the ansatz to exclude unphysical results (as a negative $\overline{v_{\varphi }^2}$). The method is illustrated by discussing the cases of three well-known galaxy models: the Miyamoto &amp; Nagai and Satoh disks, and the Binney logarithmic halo, for which the regions and the constraints on the ansatz values can be determined analytically; a two-component (Miyamoto &amp; Nagai plus logarithmic halo) model is also discussed.

Dark matter vorticity and velocity dispersion from truncated Dyson-Schwinger equations

Large-scale structure formation is studied in a kinetic theory approach, extending the standard perfect pressureless fluid description for dark matter by including the velocity dispersion tensor as a dynamical degree of freedom. The evolution of power spectra for density, velocity and velocity dispersion degrees of freedom is investigated in a non-perturbative approximation scheme based on the Dyson-Schwinger equations. In particular, the generation of vorticity and velocity dispersion is studied and predictions for the corresponding power spectra are made, which qualitatively agree well with results obtained from N-body simulations. It is found that velocity dispersion grows strongly due to non-linear effects and at late times its mean value seems to be largely independent of the initial conditions. By taking this into account, a rather realistic picture of non-linear large-scale structure formation can be obtained, albeit the numerical treatment remains challenging, especially for very cold dark matter models.

An alternative general model for the effective longitudinal diffusion in chromatographic beds filled with ordered porous particles

The two-zone moment-analysis method for the determination of the dispersion tensor in hierarchical retentive porous media has been adopted to compute and model the effective longitudinal diffusion Deff, or equivalently the B-term band broadening, in chromatographic beds filled with ordered porous particles. On the one hand, this approach offers accurate numerical results for Deff while keeping computational expenses low. On the other hand, it also gives direct insight for the analytical modelling, readily revealings the two main essential quantities (resp. referred to as the mobile-zone and stationary-zone effective diffusion factors γm and γs) that contribute to Deff. Modelling these two main parameters provided us with two new analytical models for Deff: a general one, valid for diluted and concentrated packings and accurate in the whole range of relevant intra-particle diffusion coefficient Dpz, and an approximate one, reliable for diluted packings and accurate also for concentrated packings with low to intermediate values of Dpz. The large advantage of both models is that they do not need any fitting parameter because all the required information is incorporated into the experimentally accessible geometric obstruction factor in the mobile phase originating from the tortuosity of the through-pore space (limiting case of fully solid particles without any retention). These models hence serve as an alternative to the Effective Medium Theory (EMT) models used so far in the literature. To validate the theory, five ordered geometries have been investigated. The accuracy of the general model proposed has been quantified and found to be comparable with that of the 3rd order approximate Torquato model for four geometries, even for macro-porosities close to the close-packing limit. The case of a 2-d triangular array of ellipsoidal particles with different elongations is also investigated to show the general validity and applicability of the models.

A fast and accurate method for transport and dispersion of phosphogypsum in coastal zones: Application to Jorf Lasfar

SummaryWe present a numerical method for modelling and simulation of transport and dispersion of phosphogypsum in the Jorf Lasfar coastal zone located on the Atlantic Ocean at Morocco. The governing equations consist of the well‐established barotropic ocean model including the barometric pressure, friction terms, Coriolis and wind stresses. To model transport and dispersion of phosphogypsum we consider an advection‐diffusion equation with an anisotropic dispersion tensor and source terms. As a numerical solver, we propose a novel multilevel adaptive semi‐Lagrangian finite element method. The proposed method combines the modified method of characteristics to deal with convection terms, the finite element discretization to handle complex geometries, a projection‐based algorithm to solve the Stokes problem, and an adaptive ‐projection using quadrature rules to improve the efficiency and accuracy of the method. Numerical results are presented to demonstrate the high resolution of the proposed method and to confirm its capability to provide accurate and efficient simulations for transport and dispersion of phosphogypsum in the Jorf Lasfar coastal zone.

Dispersion Versus Diffusion in Mixing Fronts

AbstractMixing fronts form when fluids with different chemical compositions are brought into contact. They influence a large range of biogeochemical processes in hydrological systems. An important mechanism governing mixing rates in such fronts is stretching by non‐uniform flows that accelerates diffusive mass transfer by enhancing concentration gradients. In a range of systems, including porous media at Darcy scale, hydrodynamic dispersion dominates over diffusion to control local mixing rates. As it differs from diffusion through its velocity‐dependent dispersion tensor, it is not known how local dispersion interacts with macroscopic mixing front stretching. Here, we investigate the impact of local dispersion versus diffusion on the properties of steady mixing fronts created by both uniform and non‐uniform flows. We derive analytical solutions for the concentration profile, mixing scale and mixing rate across the fronts. We validate these predictions by comparison with numerical simulations and experiments performed in quasi two‐dimensional tanks over a broad range of Péclet numbers. Without porous media, the mixing scale is governed by local diffusion coupled with flow: it increases diffusively along streamlines in uniform flows while it is constant in converging flows due to the balance between fluid compression and local diffusion. With porous media, the Batchelor scale is no longer sustained and the mixing scale grows with dispersion in non‐uniform flows. In addition, the coupling between flow acceleration and dispersion results in a flow rate independent mixing interface, in contrast with the local diffusion scenario. We discuss the consequences of these findings on mixing rates in mixing fronts.

Transport of a passive scalar in wide channels with surface topography: An asymptotic theory

We generalize classical dispersion theory for a passive scalar to derive an asymptotic long-time convection–diffusion equation for a solute suspended in a wide, structured channel and subject to a steady low-Reynolds-number shear flow. Our asymptotic theory relies on a domain perturbation approach for small roughness amplitudes of the channel and holds for general surface shapes expandable as a Fourier series. We determine an anisotropic dispersion tensor, which depends on the characteristic wavelengths and amplitude of the surface structure. For surfaces whose corrugations are tilted with respect to the applied flow direction, we find that dispersion along the principal direction (i.e. the principal eigenvector of the dispersion tensor) is at an angle to the main flow direction and becomes enhanced relative to classical Taylor dispersion. In contrast, dispersion perpendicular to it can decrease compared to the short-time diffusivity of the particles. Furthermore, for an arbitrary surface shape represented in terms of a Fourier decomposition, we find that each Fourier mode contributes at leading order a linearly-independent correction to the classical Taylor dispersion diffusion tensor.

A framework for assessing the 4th rank dispersivity tensor under anisotropic axial symmetries

A framework for assessing the 4th rank dispersivity tensor under anisotropic axial symmetries

Interfacial mass transfer and axial liquid dispersion in aerated column bioreactors

The upscaling of gas–liquid mass transfer and bioreaction in aerated cultures was performed within the volume averaging theory. Effective transport equations, including their associated closure problems, were derived for 1) the main substrates and biomass (non-transferable components), and 2) the gaseous substrate (transferable component). For the transferable component, two approaches were applied: 2a) the local mass equilibrium (one equation), and 2b) the two-fluid model (two equations). The nature of the dispersion tensors in the three averaged models is discussed. Numerical computations of the axial dispersion coefficient and the volumetric mass transfer coefficient were performed by solving the associated closure problems in unit cells (0.1 ≤ Re ≤ 300; 1 ≤ Sc ≤ 550; 0.1 ≤ Pe ≤ 300; Mo = 2.5 × 1011; 0.001 ≤ εG ≤ 0.05). The results of both parameters agree with trends reported in literature; some differences in orders of magnitude for DLzz are attributable to the mismatch between the experimental conditions and the occurrence of mesoscale interactions undetected in the computational cells.

Multiscale simulation of colloids ingressing porous layers with evolving internal structure

We report on a reaction-diffusion model posed on multiple spatial scales that accounts for diffusion, aggregation, fragmentation, and deposition of populations of colloidal particles. The model is able to account for the heterogeneity of the internal porous structure of the layer. For simplicity, we represent the microstructures as discs with prescribed initial random distribution of radii. As microstructures grow due to the deposition of populations of colloidal particles, local clogging becomes possible, that is neighbouring disks may touch each other. We investigate how distributions of evolving microstructures influence the transport and storage properties of porous layers. As working tool, we propose a FD-FEM discretization of the multiscale model. We illustrate numerically local clogging effects on the dispersion tensor and quantify herewith the layer’s performance with respect to both the efficiency of the transport and the storage capacity. The presented model and numerical approach can be extended in a rather straightforward way to handle slightly more complex geometrical settings like thin porous structures with multi-layers in 2D, or single layers in 3D.

Unsaturated Transport Modeling: Random‐Walk Particle‐Tracking as a Numerical‐Dispersion Free and Efficient Alternative to Eulerian Methods

AbstractLagrangian methods, such as the random‐walk particle‐tracking (RWPT), are often qualified as a potentially valuable alternative to error‐prone Eulerian methods for simulating solute transport in unsaturated porous media. Yet, the RWPT method has not yet been validated against ‐ and compared to ‐ currently used Eulerian solutions for simulating solute transport under a range of typical unsaturated conditions. This paper presents a new implementation of the RWPT approach for advective ‐ dispersive transport problems under variably saturated conditions. We first show that, as previously demonstrated for a heterogeneous dispersion tensor, using an interpolation scheme in the RWPT algorithm performs well for problems with abrupt changes in the water content. The new model is then compared against a simple 1D uniform transport problem, for which an analytical solution exist, and against a variety of 1D and 3D numerical solutions using the different Eulerian schemes implemented in Hydrus software suite. Results show that, while the Eulerian solutions significantly suffer from numerical dispersion in case of a coarse spatial discretization of the simulation domain, the new Lagrangian model provides accurate solutions for all problems. Furthermore, RWPT reproduces accurately solute transport for typical unsaturated flow conditions (infiltration, evaporation). Moreover, the Lagrangian model appears to be orders of magnitude faster than its Eulerian alternative to solve a 3D heterogeneous problem. Thus, RWPT should be seen as an attractive, stable and efficient alternative for simulating solute transport in the vadose zone, especially in case of complex and large problems.

Dispersion tensor in stratified porous media.

Dispersion in porous media is of great importance in many areas of science and engineering. While dispersion in porous media has been generally well discussed in the literature, little work has been done regarding a generalization of Taylor dispersion in stratified media. In this work, we generalized the Taylor dispersion theory and Stokes flow in porous media to derive a reduced-order model for tracer dispersion in stratified porous media. Our findings revealed that for a simple case of two-layer porous media, the hydrodynamic coupling between the two layers leads to the tensorial nature of dispersion and advection. The results showed that the obtained dispersion tensor and advection are not symmetric unless both porous layers have similar thickness, porosity, and molecular diffusion. We found that the main elements of the coefficient of the dispersion tensor remain positive while the off-diagonal elements can take negative values. On the contrary, all elements of the advection matrix may take negative values. On the basis of these observations, we report the manifestation of the dispersion barrier, uphill dispersion and advection, and osmotic dispersion during tracer transport in stratified porous media. In particular, the identified uphill advection reveals that the injected tracer in one layer could be transported countercurrent to the adjacent layer. Furthermore, we have shown that in the limiting case of Darcy flow, the Taylor dispersion is absent, and the tracer mixing between the two layers is restricted to the cross-diffusive flux between them. The results revealed that the field scale mixing may not necessarily originate from the Taylor dispersion and could be due to the modified advection terms and the cross-diffusive flux between the two layers.

Convective dissolution of carbon dioxide in two- and three-dimensional porous media: The impact of hydrodynamic dispersion

Convective dissolution is the process by which CO2 injected in geological formations dissolves into the aqueous phase and thus remains stored perennially by gravity. It can be modeled by buoyancy-coupled Darcy flow and solute transport. The transport equation should include a diffusive term accounting for hydrodynamic dispersion, wherein the effective diffusion coefficient is proportional to the local interstitial velocity. We investigate the impact of the hydrodynamic dispersion tensor on convective dissolution in two-dimensional (2D) and three-dimensional (3D) homogeneous porous media. Using a novel numerical model, we systematically analyze, among other observables, the time evolution of the fingers' structure, dissolution flux in the quasi-constant flux regime, and mean concentration of the dissolved CO2; we also determine the onset time of convection, ton. For a given Rayleigh number Ra, the efficiency of convective dissolution over long times is controlled by ton. For porous media with a dispersion anisotropy commonly found in the subsurface, ton increases as a function of the longitudinal dispersion's strength (S), in agreement with previous experimental findings and in contrast to previous numerical findings, a discrepancy that we explain. More generally, for a given strength of transverse dispersion, longitudinal dispersion always slows down convective dissolution, while for a given strength of longitudinal dispersion, transverse dispersion always accelerates it. Furthermore, a systematic comparison between 2D and 3D results shows that they are consistent on all accounts, except for a slight difference in ton and a significant impact of Ra on the dependence of the finger number density on S in 3D.

Identification of Hydrodynamic Dispersion Tensor by Optimization Algorithm Based LBM/CMA-ES Combination

The hydrodynamic dispersion tensor (HDT) of a porous medium is a key parameter in engineering and environmental sciences. Its knowledge allows for example, to accurately predict the propagation of a pollution front induced by a surface (or subsurface) flow. This paper proposes a new mathematical model based on inverse problem-solving techniques to identify the HDT (noted D=) of the studied porous medium. We then showed that in practice, this new model can be written in the form of an integrated optimization algorithm (IOA). The IOA is based on the numerical solution of the direct problem (which solves the convection–diffusion type transport equation) and the optimization of the error function between the simulated concentration field and that observed at the application site. The partial differential equations of the direct model were solved by high resolution of (Δx=Δy=1 m) Lattice Boltzmann Method (LBM) whose computational code is named HYDRODISP-LBM (HYDRO-DISpersion by LBM). As for the optimization step, we opted for the CMA-ES (Covariance Matrix Adaptation-Evolution Strategy) algorithm. Our choice for these two methods was motivated by their excellent performance proven in the abundant literature. The paper describes in detail the operation of the coupling of the two computer codes forming the IOA that we have named HYDRODISP-LBM/CMA-ES. Finally, the IOA was applied at the Beauvais experimental site to identify the HDT D=. The geological analyzes of this site showed that the tensor identified by the IOA is in perfect agreement with the characteristics of the geological formation of the site which are connected with the mixing processes of the latter.

Dispersion tensor in a two-phase flow in a slit

Taylor's approach on the dispersion phenomenon is generalized for solute transport in a two-phase laminar flow of immiscible fluids in a slit. The reduced-order models for solute transport are derived using Reynolds decomposition and averaging techniques from which the exact analytical expressions for all elements of the dispersion tensor and the matrix of coefficients of the advection term are derived. It is shown that the dispersion tensor is generally not symmetric, and the asymmetry originates from the presence of an interface between the two fluids. We also discussed conditions at which the solute transport in a two-phase laminar flow in a slit lead to dispersion barrier, osmotic dispersion, and reverse dispersion. The results provide a thorough insight into modeling solute transport across an interface/film in two-phase stratified flows and find applications in the design and optimization of microfluidic devices where two fluids flow in laminar contact.

Comparison of optical response from DFT random phase approximation and a low-energy effective model: Strained phosphorene

The engineering of the optical response of materials is a paradigm that demands microscopic-level accuracy and reliable predictive theoretical tools. Here we compare and contrast the dispersive permittivity tensor, using both a low-energy effective model and density functional theory (DFT). As a representative material, phosphorene subject to strain is considered. Employing a low-energy model Hamiltonian with a Green's function current-current correlation function, we compute the dynamical optical conductivity and its associated permittivity tensor. For the DFT approach, first-principles calculations make use of the first-order random phase approximation. Our results reveal that although the two models are generally in agreement within the low-strain and low-frequency regime, the intricate features associated with the fundamental physical properties of the system and optoelectronics devices implementation such as band gap, Drude absorption response, vanishing real part, absorptivity, and sign of permittivity over the frequency range show significant discrepancies. Our results suggest that the random phase approximation employed in widely used DFT packages should be revisited and improved to be able to predict these fundamental electronic characteristics of a given material with confidence. Furthermore, employing the permittivity results from both models, we uncover the pivotal role that phosphorene can play in optoelectronics devices to facilitate highly programable perfect absorption of electromagnetic waves by manipulating the chemical potential and exerting strain and illustrate how reliable predictions for the dielectric response of a given material are crucial to precise device design.