Elucidating the Role of Interfacial Tension for Hydrological Properties of Two-Phase Flow in Natural Sandstone by an Improved Lattice Boltzmann Method

We investigated the interfacial tension (IFT) effect on fluid flow characteristics inside micro-scale, porous media by a highly efficient multi-phase lattice Boltzmann method using a graphics processing unit. IFT is one of the most important parameters for carbon capture and storage and enhanced oil recovery. Rock pores of Berea sandstone were reconstructed from micro-CT scanned images, and multi-phase flows were simulated for the digital rock model at extremely high resolution (3.2  \(\upmu \) m). Under different IFT conditions, numerical analyses were carried out first to investigate the variation in relative permeability, and then to clarify evolution of the saturation distribution of injected fluid. We confirmed that the relative permeability decreases with increasing IFT due to growing capillary trapping intensity. It was also observed that with certain pressure gradient \(\Delta P\) two crucial IFT values, \(\sigma _{1}\) and \(\sigma _{2}\) , exist, creating three zones in which the displacement process has totally different characteristics. When \(\sigma _{1}< \sigma < \sigma _{2}\) , the capillary fingering patterns are observed, while for \(\sigma < \sigma _{1}\) viscous fingering is dominant and most of the passable pore spaces were invaded. When \(\sigma > \sigma _{2}\) the invading fluid failed to break through. The pore-throat-size distribution estimated from these crucial IFT values ( \(\sigma _{1 }\) and \(\sigma _{2})\) agrees with that derived from mercury porosimetry measurements of Berea sandstone. This study demonstrates that the proposed numerical method is an efficient tool for investigating hydrological properties from pore structures.     

Experimental Characterization of Porosity Structure and Transport Property Changes in Limestone Undergoing Different Dissolution Regimes

Limestone dissolution by $\hbox {CO}_2$ -rich brine induces critical changes of the pore network geometrical parameters such as the pore size distribution, the connectivity, and the tortuosity which govern the macroscopic transport properties (permeability and dispersivity) that are required to parameterize the models, simulating the injection and the fate of $\hbox {CO}_2$ . A set of four reactive core-flood experiments reproducing underground conditions ( $T = 100\,^{\circ }\hbox {C}$ and $P = 12$ MPa) has been conducted for different $\hbox {CO}_2$ partial pressures $(0.034 < P_{\mathrm{CO}_2}< 3.4\; \hbox {MPa})$ in order to study the different dissolution regimes. X-ray microtomographic images have been used to characterize the changes in the structural properties from pore scale to Darcy scale, while time-resolved pressure loss and chemical fluxes enabled the determination of the sample-scale change in porosity and permeability. The results show the growth of localized dissolution features associated with high permeability increase for the highest $P_{\mathrm{CO}_2}$ , whereas dissolution tends to be more homogeneously distributed for lower values of $P_{\mathrm{CO}_2}$ . For the latter, the higher the $P_{\mathrm{CO}_2}$ , the more the dissolution patterns display ramified structures and permeability increase. For the lowest value of $P_{\mathrm{CO}_2}$ , the preferential dissolution of the calcite cement associated with the low dissolution kinetics triggers the transport that may locally accumulate and form a microporous material that alters permeability and produces an anti-correlated porosity–permeability relationship. The combined analysis of the pore network geometry and the macroscopic measurements shows that $P_{\mathrm{CO}_2}$ regulates the tortuosity change during dissolution. Conversely, the increase of the exponent value of the observed power law permeability–porosity trend while $P_{\mathrm{CO}_2}$ increases, which appears to be strongly linked to the increase of the effective hydraulic diameter, depends on the initial rock structure.     

Correlation of Water Vapor Permeability with Microstructure Characteristics of Building Materials Using Robust Chemometrics

This paper studies various microstructure parameters of natural and artificial building materials and aims to their correlation to the water vapor permeability. Three categories of building materials were investigated: stones, bricks, and plasters. Mercury intrusion porosimetry was applied in order to obtain the materials microstructure characteristics, a variety of pore size distributions and pore structure measurements, such as total porosity. The water vapor permeability of materials was determined experimentally according to ASTM standard E96-00. A robust principal component regression approach, coupled with multiple outlier detection, was applied in order to correlate water vapor permeability values to pore size distributions. A good quality correlation model was found by utilizing relative specific pore volume and relative specific pore surface distributions, whereas using pore structure measurements, such as total porosity, the correlation results were very poor. From the results, specific ranges of pore size distribution, corresponding to pores radius sizes greater than $10\,\upmu \text{ m }$ 10 μ m and between 1.778 and $0.421\,\upmu \text{ m }$ 0.421 μ m , contribute to the water vapor permeability of the materials.     

Impact of Effective Stress and Matrix Deformation on the Coal Fracture Permeability

The permeability of coal is an important parameter in mine methane control and coal bed methane exploitation because it determines the practicability of methane extraction. We developed a new coal permeability model under tri-axial stress conditions. In our model, the coal matrix is compressible and Biot’s coefficient, which is considered to be 1 in existing models, varies between 0 and 1. Only a portion of the matrix deformation, which is represented by the effective coal matrix deformation factor $f_\mathrm{m}$ , contributes to fracture deformation. The factor $f_\mathrm{m}$ is a parameter of the coal structure and is a constant between 0 and 1 for a specific coal. Laboratory tests indicate that the Sulcis coal sample has an $f_\mathrm{m}$ value of 0.1794 for $\hbox {N}_{2}$ and $\hbox {CO}_{2}$ . The proposed permeability model was evaluated using published data for the Sulcis coal sample and is compared to three popular permeability models. The proposed model agrees well with the observed permeability changes and can predict the permeability of coal better than the other models. The sensitivity of the new model to changes in the physical, mechanical and adsorption deformation parameters of the coal was investigated. Biot’s coefficient and the bulk modulus mainly affect the effective stress term in the proposed model. The sorption deformation parameters and the factor $f_\mathrm{m}$ affect the coal matrix deformation term.     

Drainage of Capillary-Trapped Oil by an Immiscible Gas: Impact of Transient and Steady-State Water Displacement on Three-Phase Oil Permeability

In a previous paper (Dehghanpour et al., Phys Rev E 83:065302, 2011a), we showed that relative permeability of mobilized oil, $k_\mathrm{ro}$ , measured during tertiary gravity drainage, is significantly higher than that of the same oil saturation in other tests where oil is initially a continuous phase. We also showed that tertiary $k_\mathrm{ro}$ strongly correlates to both water saturation, $S_\mathrm{w}$ , water flux (water relative permeability), $k_\mathrm{rw}$ , and the change in water saturation with time, $\mathrm{d}S_\mathrm{w}/\mathrm{d}t$ . To develop a model and understanding of the enhanced oil transport, identifying which of these parameters ( $S_\mathrm{w},\,k_{\mathrm{rw}}$ , or $\mathrm{d}S_\mathrm{w}/\mathrm{d}t$ ) plays the controlling role is necessary, but in the previous experiments these could not be deconvolved. To answer the remaining question, we conduct specific three-phase displacement experiments in which $k_{\mathrm{rw}}$ is controlled by applying a fixed water influx, and $S_\mathrm{w}$ develops naturally. We obtain $k_{\mathrm{ro}}$ by using the saturation data measured in time and space. The results suggest that steady-state water influx, in contrast to transient water displacement, does not enhance $k_{\mathrm{ro}}$ . Instead, reducing water influx rate results in excess oil flow. Furthermore, according to our pore scale hydraulic conductivity calculations, viscous coupling and fluid positioning do not sufficiently explain the observed correlation between $k_{\mathrm{ro}}$ and $S_{\mathrm{w}}$ . We conclude that tertiary $k_{\mathrm{ro}}$ is controlled by the oil mobilization rate, which in turn is linked to the rate of water saturation decrease with time, $\mathrm{d}S_\mathrm{w}/\mathrm{d}t$ . Finally, we develop a simple model which relates tertiary $k_{\mathrm{ro}}$ to transient two-phase gas/water relative permeability.     

Micro-structural Impact of Different Strut Shapes and Porosity on Hydraulic Properties of Kelvin-Like Metal Foams

Various ideal periodic isotropic structures of foams (tetrakaidecahedron) with constant ligament cross section are studied. Different strut shapes namely circular, square, diamond, hexagon, star, and their various orientations are modeled using CAD. We performed direct numerical simulations at pore scale, solving Navier–Stokes equation in the fluid space to obtain various flow properties namely permeability and inertia coefficient for all shapes in the porosity range, \(0.60<\varepsilon <0.95\) for wide range of Reynolds numbers, \(10^{-6} . We proposed an analytical model to obtain pressure drop in metallic foams in order to correlate the resulting macroscopic pressure and velocity gradients with the Ergun-like approach. The analytical results are fully compared with the available numerical data, and an excellent agreement is observed.     

On the Potential of Tomographic Methods when Applied to Compacted Crushed Rock Salt

Compacted crushed rock salt is considered as potential backfill material in repositories for nuclear waste. To evaluate the sealing properties of this material knowledge concerning the nature of the pore space is of eminent interest. Here, the pore microstructures of crushed rock salt samples with different compaction states were investigated by X-ray (XCT) computed tomography and Focused Ion Beam nanotomography (FIB-nt). Based on these methods the pore microstructures were reconstructed and quantitatively analyzed with respect to porosity, connectivity and percolation properties. Regarding pores with radii \(> 4\,\upmu \hbox {m}\) , porosity differs substantially in the two analyzed samples ( \(\phi = 0.01\) and 0.10). The pore microstructures are considered isotropic in connectivity and percolation threshold. Using two finite-scaling schemes we found percolation thresholds with critical porosities \(\phi _{c} > 0.05\) . Based on statistical considerations, the millimeter size samples that can be analyzed by XCT are large enough to provide a meaningful picture of the pore geometry related to macroporosity. The samples contain also a small fraction (i.e. \(< 0.01\) ) of pores with radii \(< 1\,\upmu \hbox {m}\) , which were resolved by FIB-nt. Often these pores can be found along grain boundaries. These pores are granular shaped and are not connected to each other. Typical samples size that can be analyzed by FIB-nt is on the order of tens of microns, which turned out to be too small to provide representative geometric information unless an effort is made that involves several FIB-nt realizations per sample.     

Necessary and Sufficient Conditions for the Invertibility of the Nonlinear Difference Operator \left( {\mathcal{D}x} \right)\left( t \right) = x\left( {t + 1}\right) - f\left( {x\left( t \right)} \right) in the Space of Functions Bounded and Continuous on the Axis

We find necessary and sufficient conditions for the nonlinear difference operator $\left( {\mathcal{D}x} \right)\left( t \right) = x\left( {t + 1} \right) - f\left( {x\left( t \right)} \right)$ $t \in \mathbb{R}$ , where $f:\mathbb{R} \to \mathbb{R}$ is a continuous function, to have the inverse in the space of functions bounded and continuous on $\mathbb{R}$ .     

Pore Space Microstructure Evolution of Regular Sphere Packings Undergoing Compaction and Cementation

We examine the pore space structure evolution of ordered uniform sphere packs: simple cubic (SC), body centered cubic (BCC), and face centered cubic (FCC), undergoing simple diagenetic processes that reduce their pore spaces. Focus is on the occurrence of pore space microstructure changes or transitions, which are followed through their characteristic or critical pore lengths (l _c). For almost all the cubic packings undergoing either compaction or cementation there are no singularities in l _c. This is a consequence of having a single pore shape controlling flow at all stages of the process. However, this is not so for the BCC packing under cementation, for which l _c is non-monotonic exhibiting a kink at ${\phi \approx 0.1452}$ , the porosity at which the pore shape controlling flow switches to a different form and position. These results for uniform compaction/cementation complement our previous works on pore networks under random shrinkage. Kinks in l _c as porosity decreases signal pore space microstructure transitions that anticipate sudden changes in the permeability?Cporosity relation as porosity decreases. The consequences are great; clearly l _c is not a constant unless the diagenetic process is mild. A l _c function of compaction/cementation advancement should be used above a transition and a different l _c function below. For the sphere packs here, once the diagenetic process has reduced the pore space substantially, a l _c function of compaction/cementation advancement is mandatory if we are to capture all significant flow features.     

Tensor Products, Positive Linear Operators, and Delay-Differential Equations

We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equation $$\begin{aligned} \dot{x}(t)=-\alpha (t)x(t)-\beta (t)x(t-1) \end{aligned}$$ with a single delay, where the delay coefficient is of one sign, say $\delta \beta (t)\ge 0$ with $\delta \in \{-1,1\}$ . Positivity properties are studied, with the result that if $(-1)^k=\delta $ then the $k$ -fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients $\alpha (t)$ and $\beta (t)$ are periodic of the same period, and $\beta (t)$ satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of $u_0$ -positivity of the exterior product is investigated when $\beta (t)$ satisfies a uniform sign condition.     

Statistical Scaling of Geometric Characteristics in Millimeter Scale Natural Porous Media

We analyze statistical scaling of structural attributes of two millimeter scale rock samples, Estaillades limestone and Bentheimer sandstone. The two samples have different connected porosities and pore structures. The pore-space geometry of each sample is reconstructed via X-ray micro-tomography at micrometer resolution. Directional distributions of porosity and specific surface area (SSA), which are key Minkowski functionals (geometric observables) employed to describe the pore-space structure, are calculated from the images, and scaling of associated order- $q$ sample structure functions of absolute incremental values is analyzed. Increments of porosity and SSA tend to be statistically dependent and persistent (tendency for large and small values to alternate mildly) in space. Structure functions scale as powers $\xi (q)$ of directional separation distance or lag, $s$ , over an intermediate range of $s$ , displaying breakdown in power law scaling at large and small lags. Powers $\xi \!\!\left( q \right) $ of porosity and SSA inferred from moment and extended self-similarity (ESS) analyses of limestone and sandstone data tend to be quasi-linear and nonlinear (concave) in $q$ , respectively. We observe an anisotropic behavior for $\xi (q)$ , which appears to be mild for the porosity of the sandstone sample while it is marked for both porosity and SSA of the limestone rock sample. The documented nonlinear scaling behavior is amenable to analysis by viewing the variables as samples from sub-Gaussian random fields subordinated to truncated fractional Brownian motion or fractional Gaussian noise.     

A Procedure for the Accurate Determination of Sub-Core Scale Permeability Distributions with Error Quantification

We present a new method for non-destructively calculating sub-core scale permeability distributions within a core. The new method integrates experimentally measured capillary pressure data and sub-core scale saturation and porosity data collected using a computed tomography-scanner, to construct an accurate and unique sub-core scale permeability distribution. Using this procedure, it is possible to conduct highly refined simulations of core flooding experiments without typical assumptions requiring the core to be homogeneous, or relying on inaccurate porosity-based methods for estimating permeability distributions. The calculation procedure is described and results from two example rock cores are presented, a Berea Sandstone and a sandstone from the Otway Basin Pilot Project in Australia. Drainage coreflooding experiments of carbon dioxide ( $\text{ CO }_{2})$ CO 2 ) injection into water are first conducted on both cores and permeability distributions are calculated using the experimental data. Numerical simulations of the very same experiments are then conducted to demonstrate the accuracy of the calculated sub-core scale permeability distribution. Results from both cores show that the input sub-core scale saturation distributions are predicted with an $R^{2}$ R 2 correlation of greater than 0.93. This is compared to having no correlation when using simple porosity-only based permeability distributions, or assuming homogeneous core properties (Krause et al., SPE J 16(4):768–777, 2011). The uniqueness of the calculated permeability distribution is then demonstrated by calculating permeability distributions for the same core using data collected at different $\text{ CO }_{2}$ CO 2 injection fractional flows. Results show that the two independently calculated permeability distributions agree within the limits of experimental measurement error.     

Conjugate Natural Convection in a Porous Enclosure Sandwiched by Finite Walls Under the Influence of Non-uniform Heat Generation and Radiation

Conjugate natural convection in a square porous enclosure sandwiched by finite walls under the influence of non-uniform heat generation and radiation is studied numerically in the present article. The horizontal heating is considered, where the vertical walls heated isothermally at different temperatures, while the horizontal walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and finite difference method is applied to solve the dimensionless governing equations. The governing parameters considered are the ratio of wall thickness to its width $(0.02 \le D \le 0.3)$ ( 0.02 ≤ D ≤ 0.3 ) , the wall to porous thermal conductivity ratio $(0.1 \le k_\mathrm{r} \le 10.0)$ ( 0.1 ≤ k r ≤ 10.0 ) , the internal heating $(0 \le \gamma \le 5)$ ( 0 ≤ γ ≤ 5 ) , and the local heating exponent parameters $(1 \le \lambda \le 20)$ ( 1 ≤ λ ≤ 20 ) . It is found that the average Nusselt number on the hot and cold interfaces increases with increasing the radiation intensity. Very high non-uniformity heating does not affect the average Nusselt number at very thick walls.     

Generalized Flows Satisfying Spatial Boundary Conditions

In a region D in ${\mathbb{R}^2}$ or ${\mathbb{R}^3}$ , the classical Euler equation for the regular motion of an inviscid and incompressible fluid of constant density is given by $$\partial_t v+(v\cdot \nabla_x)v=-\nabla_x p, {\rm div}_x v=0,$$ where v(t, x) is the velocity of the particle located at ${x\in D}$ at time t and ${p(t,x)\in\mathbb{R}}$ is the pressure. Solutions v and p to the Euler equation can be obtained by solving $$\left\{\begin{array}{l} \nabla_x\left\{\partial_t\phi(t,x,a) + p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2 \right\}=0\,{\rm at}\,a=\kappa(t,x),\\ v(t,x)=\nabla_x \phi(t,x,a)\,{\rm at}\,a=\kappa(t,x), \\ \partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, \\ {\rm div}_x v(t,x)=0, \end{array}\right. \quad\quad\quad\quad\quad(0.1)$$ where $$\phi:\mathbb{R}\times D\times \mathbb{R}^l\rightarrow\mathbb{R}\,{\rm and}\, \kappa:\mathbb{R}\times D \rightarrow \mathbb{R}^l$$ are additional unknown mappings (l?≥ 1 is prescribed). The third equation in the system says that ${\kappa\in\mathbb{R}^l}$ is convected by the flow and the second one that ${\phi}$ can be interpreted as some kind of velocity potential. However vorticity is not precluded thanks to the dependence on a. With the additional condition κ(0, x)?=?x on D (and thus l?=?2 or 3), this formulation was developed by Brenier (Commun Pure Appl Math 52:411–452, 1999) in his Eulerian–Lagrangian variational approach to the Euler equation. He considered generalized flows that do not cross ${\partial D}$ and that carry each “particle” at time t?=?0 at a prescribed location at time t?=?T?>?0, that is, κ(T, x) is prescribed in D for all ${x\in D}$ . We are concerned with flows that are periodic in time and with prescribed flux through each point of the boundary ${\partial D}$ of the bounded region D (a two- or three-dimensional straight pipe). More precisely, the boundary condition is on the flux through ${\partial D}$ of particles labelled by each value of κ at each point of ${\partial D}$ . One of the main novelties is the introduction of a prescribed “generalized” Bernoulli’s function ${H:\mathbb{R}^l\rightarrow \mathbb{R}}$ , namely, we add to (0.1) the requirement that $$\partial_t\phi(t,x,a) +p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2=H(a)\,{\rm at}\,a=\kappa(t,x)\quad\quad\quad\quad\quad(0.2)$$ with ${\phi,p,\kappa}$ periodic in time of prescribed period T?>?0. Equations (0.1) and (0.2) have a geometrical interpretation that is related to the notions of “Lamb’s surfaces” and “isotropic manifolds” in symplectic geometry. They may lead to flows with vorticity. An important advantage of Brenier’s formulation and its present adaptation consists in the fact that, under natural hypotheses, a solution in some weak sense always exists (if the boundary conditions are not contradictory). It is found by considering the functional $$(\kappa,v)\rightarrow \int\limits_{0}^T \int\limits_D\left\{\frac 1 2 |v(t,x)|^2+H(\kappa(t,x))\right\}dt\, dx$$ defined for κ and v that are T-periodic in t, such that $$\partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, {\rm div}_x v(t,x)=0,$$ and such that they satisfy the boundary conditions. The domain of this functional is enlarged to some set of vector measures and then a minimizer can be obtained. For stationary planar flows, the approach is compared with the following standard minimization method: to minimize $$\int\limits_{]0,L[\times]0,1[} \{(1/2)|\nabla \psi|^2+H(\psi)\}dx\,{\rm for}\,\psi\in W^{1,2}(]0,L[\times]0,1[)$$ under appropriate boundary conditions, where ψ is the stream function. For a minimizer, corresponding functions ${\phi}$ and κ are given in terms of the stream function ψ.     

A Model for Spontaneous Imbibition as a Mechanism for Oil Recovery in Fractured Reservoirs

The flow of oil and water in naturally fractured reservoirs (NFR) can be highly complex and a simplified model is presented to illustrate some main features of this flow system. NFRs typically consist of low-permeable matrix rock containing a high-permeable fracture network. The effect of this network is that the advective flow bypasses the main portions of the reservoir where the oil is contained. Instead capillary forces and gravity forces are important for recovering the oil from these sections. We consider a linear fracture which is symmetrically surrounded by porous matrix. Advective flow occurs only along the fracture, while capillary driven flow occurs only along the axis of the matrix normal to the fracture. For a given set of relative permeability and capillary pressure curves, the behavior of the system is completely determined by the choice of two dimensionless parameters: (i) the ratio of time scales for advective flow in fracture to capillary flow in matrix $\alpha =\tau ^f/\tau ^m$ ; (ii) the ratio of pore volumes in matrix and fracture $\beta =V^m/V^f$ . A characteristic property of the flow in the coupled fracture–matrix medium is the linear recovery curve (before water breakthrough) which has been referred to as the “filling fracture” regime Rangel-German and Kovscek (J Pet Sci Eng 36:45–60, 2002), followed by a nonlinear period, referred to as the “instantly filled” regime, where the rate is approximately linear with the square root of time. We derive an analytical solution for the limiting case where the time scale $\tau ^{m}$ of the matrix imbibition becomes small relative to the time scale $\tau ^{f}$ of the fracture flow (i.e., $\alpha \rightarrow \infty $ ), and verify by numerical experiments that the model will converge to this limit as $\alpha $ becomes large. The model provides insight into the role played by parameters like saturation functions, injection rate, volume of fractures versus volume of matrix, different viscosity relations, and strength of capillary forces versus injection rate. Especially, a scaling number $\omega $ is suggested that seems to incorporate variations in these parameters. An interesting observation is that at $\omega =1$ there is little to gain in efficiency by reducing the injection rate. The model can be used as a tool for interpretation of laboratory experiments involving fracture–matrix flow as well as a tool for testing different transfer functions that have been suggested to use in reservoir simulators.     

A note on the soil-water conductivity of a fractal soil

We show that for a fractal soil the soil-water conductivity, K, is given by $$\frac{K}{{K_\varepsilon }} = (\Theta /\varepsilon )^{2D/3 + 2/(3 - D)}$$ where $K_\varepsilon$ is the saturated conductivity, θ the water content, ? its saturated value and D is the fractal dimension obtained from reinterpreting Millington and Quirk's equation for practical values of the porosity ?, as $$D = 2 + 3\frac{{\varepsilon ^{4/3} + (1 - \varepsilon )^{2/3} - 1}}{{2\varepsilon ^{4/3} \ln ,{\text{ }}\varepsilon ^{ - 1} + (1 - \varepsilon )^{2/3} \ln (1 - \varepsilon )^{ - 1} }}$$ .     

Predominant Gas Transport in Microporous Hydrotalcite–Silica Membrane

The prepared microporous hydrotalcite (HT)–silica membrane was found to exhibit the molecular sieving characteristic of pristine silica material and high $\mathrm{CO}_{2}$ adsorption capacity of HT. The combined properties made enhanced $\mathrm{CO}_{2}$ permeability and separability from $\mathrm{CH}_{4}$ possible. The gas transport in the membrane was predominantly surface adsorption. The porous membrane overcame the Knudsen limitation and yielded the highest separation selectivity of 120 at 40 % $\mathrm{CO}_{2}$ feed concentration, $30\,^{\circ }\mathrm{C}$ operating temperature, and 100 kPa pressure difference.     

Singular Perturbation of Reduced Wave Equation and Scattering from an Embedded Obstacle

We consider time-harmonic wave scattering from an inhomogeneous isotropic medium supported in a bounded domain ${\Omega \subset \mathbb{R}^N}$ (N ≥?2). In a subregion ${D \Subset \Omega}$ , the medium is supposed to be lossy and have a large mass density. We study the asymptotic development of the wave field as the mass density ρ → +?∞ and show that the wave field inside D will decay exponentially while the wave filed outside the medium will converge to the one corresponding to a sound-hard obstacle ${D \Subset \Omega}$ buried in the medium supported in ${\Omega \backslash \overline{D}}$ . Moreover, the normal velocity of the wave field on ? D from outside D is shown to be vanishing as ρ → +?∞. We derive very accurate estimates for the wave field inside and outside D and on ? D in terms of ρ, and show that the asymptotic estimates are sharp. The implication of the obtained results is given for an inverse scattering problem of reconstructing a complex scatterer.     

Double-Diffusive Natural Convection in Anisotropic Porous Medium Bounded by Finite Thickness Walls: Validity of Local Thermal Equilibrium Assumption

Double-diffusive natural convection in fluid-saturated porous medium inside a vertical enclosure bounded by finite thickness walls with opposing temperature, concentration gradients on vertical walls as well as adiabatic and impermeable horizontal ones has been performed numerically. The Darcy model was used to predict fluid flow inside the porous material, while thermal fields are simulated based on two-energy equations for fluid and solid phases on the basis of a local thermal non-equilibrium model. Computations have been performed for different controlling parameters such as the buoyancy ratio $N$ , the Lewis number Le, the anisotropic permeability ratio $R_\mathrm{p}$ , the fluid-to-solid thermal conductivity ratio $R_\mathrm{c}$ , the interphase heat transfer coefficient $\mathcal{H}$ , the ratio of the wall thickness to its height $D$ , the wall-to-porous medium thermal diffusivity ratio $R_\mathrm{w}$ , and the solid-to-fluid heat capacity ratio $\gamma $ . Thus, the effects of the controlling parameters on heat and mass transfer characteristics are discussed in detail. Moreover, the validity domain of the local thermal equilibrium (LTE) assumption has been delimited for different set of the governing parameters. It has been shown that Le has a noticeable significant effect on fluid temperature profiles and that higher $N$ values lead to a significant enhancement in heat and mass transfer rates. Moreover, for higher $\mathcal{H}, R_\mathrm{c}$ , $R_\mathrm{p}, R_\mathrm{w}$ , or $D$ values and/or lower $\gamma $ values, the solid and fluid phases tend toward LTE.     

Extension of k–ε models for a turbulent falling film

An extension applicable to various k–ε models is proposed to account for the damping of turbulence due to the surface tension. It consists of a sink in the equation for $ {{\overline{D} k} \mathord{\left/ {\vphantom {{\overline{D} k} {\overline{D} t}}} \right. \kern-0em} {\overline{D} t}} $ and a source in the equation for $ {{\overline{D} \varepsilon } \mathord{\left/ {\vphantom {{\overline{D} \varepsilon } {\overline{D} t}}} \right. \kern-0em} {\overline{D} t}} $ both derived from dimensional analysis. First numerical experiments are undertaken with a commercial CFD software. Reasons are given why the tuning of the model should be performed with thermal experiments. The proposed model is practical, since it only needs the programming of source terms. On account of it’s mathematical structure it needs a very small closure coefficient. The intention of this article is to stimulate the numerical turbulence research in a way that is applicable to the engineering practice. A comparison to experimental data is not done here.