Linear Pressure Distribution Research Articles

The enriched-embedded discrete fracture model (nEDFM) for fluid flow in fractured porous media

The embedded discrete fracture model (EDFM) is widely adopted for simulating fluid flow in fractured porous media, but it faces limitations when dealing with blocking fractures due to its assumption of symmetric and linear pressure distribution around fractures. To improve this, our work presents a novel numerical framework named the enriched-embedded discrete fracture model (nEDFM). This framework, based on a non-conforming grid and discretized using the mass-conservative finite volume method, offers significant advancements. An attractive feature of the nEDFM is its applicability to both conductive and blocking fractures, as well as isotropic and anisotropic matrix permeability, all while maintaining a low computational cost. The local shape function, which introduces two enriched degrees of freedom for every fracture cell, is developed to model the discontinuities of pressure and its gradient on two sides of the fracture. Additionally, two equations for coupling the matrix and reduced fracture are supplemented to close the system and improve the accuracy of mass exchange between the matrix and fracture. Comparative assessments between the nEDFM, classical EDFM, and projection-based EDFM (pEDFM) demonstrate the superior performance of nEDFM. Specifically, the nEDFM outperforms the EDFM for conductive fractures and can obtain better pressure distribution inside blocking fractures than the pEDFM. In scenarios involving anisotropic matrix permeability with either conductive or blocking fractures, the nEDFM can get solutions comparable to those obtained by the reference solution. Hence, the nEDFM emerges as an accurate and flexible method for simulating fluid flow in fractured porous media.

Structural Behavior of Water Flow Glazing: Stress and Elastic Deformation Considering Hydrostatic Pressure

The fluid inside a Water Flow Glazing (WFG) panel creates a linear pressure distribution along the vertical dimension. Tension stress can cause problems with the sealant; compression stress can cause deflections in the glass panel. Increasing the glass thickness until the deflection is below the required limit implies more weight and cost of the glazing. Another solution is to limit glass deflection by inserting pillars or stripes into the cavity between two glass panes. The novelty of this article was to test a High-Order Finite Difference Method for linear and non-linear models to evaluate the effect of hydrostatic pressure produced by the fluid chamber on WFG panels. The methodology was tested on two case studies to assess the tension and deflection of the glass panes to guarantee the structural stability of WFG. The main conclusion drawn was that a linear plate model was sufficient to dimension the width, length, and thickness of the WFG panel. Furthermore, the mathematical model provided criteria to keep the glass tension below 45 MPa for tempered glass and the maximum deflection as the minimum between 1‰ of the glass height and 10% of the water chamber thickness. Introducing pillars or stripes solved hydrostatic pressure problems when the panel’s height was above 1.5 m.

Local Embedded Discrete Fracture Model (LEDFM)

The study of flow in fractured porous media is a key ingredient for many geoscience applications, such as reservoir management and geothermal energy production. Modelling and simulation of these highly heterogeneous and geometrically complex systems require the adoption of non-standard numerical schemes. The Embedded Discrete Fracture Model (EDFM) is a simple and effective way to account for fractures with coarse and regular grids, but it suffers from some limitations: it assumes a linear pressure distribution around fractures, which holds true only far from the tips and fracture intersections, and it can be employed for highly permeable fractures only. In this paper we propose an improvement of EDFM which aims at overcoming these limitations computing an improved coupling between fractures and the surrounding porous medium by (a) relaxing the linear pressure distribution assumption, (b) accounting for impermeable fractures modifying near-fracture transmissibilities. These results are achieved by solving different types of local problems with a fine conforming grid, and computing new transmissibilities (for connections between fractures and the surrounding porous medium and those through the porous medium itself near to the fractures). Such local problems are inspired from numerical upscaling techniques present in the literature. The new method is called Local Embedded Discrete Fracture Model (LEDFM) and the results obtained from several numerical tests confirm the aforementioned improvements.

Three-Dimensional Face Stability of Earth Pressure Balance Shield Tunnels Considering Non-Uniform Support and Semi-Open Driving Mode under Seepage Conditions

This work is motivated to study the face stability of earth pressure balance shield tunnels excavated in the semi-open mode and supported by non-uniform chamber pressure under seepage conditions. Within the kinematic approach of limit analysis, an analytical framework incorporating the seepage effect and linear chamber pressure distribution is developed. The pore pressure distributions are numerically simulated and imported into the three-dimensional (3D) discrete rotational failure mechanism. The work rates caused by the pore pressure and non-uniform support in the semi-open mode are introduced into the work rate balance equation to optimize the critical support pressure. Then, a parametric analysis of three typical soils, including the sand, silt, and clay, is performed to investigate the difference between the semi-open mode and closed mode under seepage conditions. The effects of the muck height and pressure gradient in the chamber, the groundwater levels, and anisotropic permeability are also studied. Finally, several design charts of the normalized critical support are presented. The results show that the difference between considering a linear pressure distribution and a uniform distribution in the semi-open mode is small, with a maximum difference of 3%. Compared to the closed mode, the instability of the tunnel face is significantly higher in the semi-open mode under seepage conditions.

Prediction model of air infiltration in single-zone buildings with high airtightness

Air infiltration through building envelopes has a considerable impact on the comprehensive performance of buildings, especially in terms of their energy demand and indoor air quality. Therefore, it is important to accurately predict building air infiltration rates under various scenarios. High airtightness is one of the typical characteristics of passive ultra-low energy buildings. With the rapid application of passive technology in building energy efficiency, the airtightness of new urban buildings has been significantly improved. The centralized air leakage path distribution assumption of current prediction model for building air infiltration rate is inconsistent with the actual situation of high airtightness buildings, which reduces its prediction accuracy and application range. Therefore, it is of great practical significance and academic value to carry out the research on the prediction model of air infiltration rate of buildings with high airtightness.This paper presents an air infiltration prediction model for single-zone buildings with adventitious openings. The building envelope was broken down into permeable parts and impermeable parts, and the air leakage pathways were assumed to be uniformly and continuously distributed in the permeable envelope. A linear pressure distribution over the building facade was assumed, and the airflow rate was integrated in the vertical and horizontal planes to theoretically predict the air infiltration rate. The feasibility of the proposed model was tested by comparing the air infiltration rates simulated by this model with those determined using the tracer gas attenuation method of an airtight building. The initial test results suggest that this model is mathematically robust and is capable of modeling the air infiltration of a building in a wide variety of scenarios. Reasonable agreement was found between the tested and simulated results. This study can provide basic theoretical support for the coupling performance analysis of high airtightness buildings.

A finite differences approach for fluid-structure interaction in a hydrostatic compliant thrust bearing

It is well known since the pioneering work of Dowson and Castelli that load capacity increases in compliant (elastic pad) hydrostatic bearings, compared to their rigid counterparts. In a recently published paper, a new design solution of a flexible pad hydrostatic bearing was proposed and analysed theoretically. A simplified analytical model was proposed for the fluid-structure interaction. The model was based on two important simplifications: linear pressure distribution on the bearing land, used to calculate the deformation of the pad, and parabolic variation of film thickness in the Reynolds equation. The analysis was made for isothermal conditions and constant pressure in the recess. This new approach relaxes the two simplifications by considering a linearly elastic model for the flexible pad and including its deformation in the Reynolds equation. A finite differences scheme is proposed for both the fluid flow and elastic deformation, considering the flexible plate simply supported. The consistency and convergence of the iterative solution are analysed, and comparisons with previous analytical solutions are shown. These results show the limits of the accuracy of the simplified analytical model and allow the selection of the design parameters of a test rig.

Unsteady-state brush theory

This paper deals with unsteady-state brush tyre models. Starting from tyre-road contact theory, we provide a full analytical solution to the partial differential equations (PDEs) describing the bristle deformation in the adhesion region of the contact patch. We show that the latter can be divided in two different regions, corresponding to two different domains for the solution of the governing PDEs of the system. In the case of constant sliding speed inputs, the steady-state solution coincides with the one provided by the classic steady-state brush theory. For a rectangular contact patch and parabolic pressure distribution, the time trend of the shear stresses is investigated. For the pure interactions (longitudinal, lateral and camber), some important conclusions are drawn about the relaxation length. Finally, an approach to derive simplified formulae for the tangential forces arising in the contact patch is introduced; the tyre formulae obtained by using the proposed approach are not based on the common slip definition, and can be employed when the rolling speed approaches zero. The outlined procedure is applied to the cases of linear tyre forces and parabolic pressure distribution.

Preliminary Results for Flow Boiling Heat Transfer of R1233zd in Microchannels

The paper reports preliminary measurement results for flow boiling heat transfer of refrigerant R1233zd in parallel horizontal microchannels. The aluminum test section consists of two blocks and has 10 parallel channels with 1.0 mm in width, 1.5 mm in height and 440 mm in length. Five small thermocouple holes with diameter 0.6 mm and depth 20 mm were drilled at 10 locations along the channel on the two aluminum blocks, respectively. Local heat flux and channel surface temperature along the channel were obtained using the inverse heat conduction method [1] based on temperatures accurately measured by 100 thermocouples. Local saturation temperatures were obtained by assuming linear pressure distribution over the two-phase flow region along the channel. Experiments were conducted under different conditions. The refrigerant mass flow rate varied from 100 kg/m²s to 500 kg/ m²s. The heating water mass flow rate varied from 200 ml/min to 1300 ml/min. The heating water inlet temperature varied from 40℃ to 80℃. Preliminary results are presented and discussed.

Analytical solution for one-dimensional contaminant diffusion through unsaturated soils beneath geomembrane

Diffusion is the dominating process in the migration of contaminants through landfill liners, below which the soils are generally unsaturated. Analytical method is often considered as a preferred approach over numerical method to solve diffusion problems. In this study, a one-dimensional diffusion model of contaminants is firstly developed to account for unsaturated properties of the soils beneath landfill barriers. Based on an assumption of linear water pressure distribution, the nonlinear governing equation in the diffusion model is transformed into a linear equation using an exponential expression of soil water retention curve and a linear model of unsaturated diffusion coefficient. Subsequently, the corresponding analytical solution is obtained and the solution is then applied to investigate the effects of soil type, water pressure condition and groundwater depth on the diffusion of contaminants. The analytical results show that groundwater depth has the most influential effect on the steady flux rate of contaminants. Furthermore, the obtained steady flux rates in soil layers with high desaturation rate are negligible when the groundwater depth exceeds 3 m and 2 m. The results also indicate that a thicker soil layer with high desaturation rate is recommended to be an effective diffusion barrier to mitigate the pollution of the underlying soils and groundwater.

Analytical solution to heat transfer in compressible laminar flow in a flat minichannel

Heat transfer in compressible laminar flow in mini-/micro-channels, a classical and general topic in fields of fuel cells, electronics, micro heat exchanger, etc., is revisited. Based on a two-dimensional continuum flow model, analytical solutions of the dimensionless model are achieved in closed-form symbolic algebras of Whittaker eigenfunctions, corresponding to two kinds of boundary conditions with arbitrarily prescribed wall temperature or wall heat flux. As the eigenvalues and eigenfunctions are independent on the dimensionless quantities, which influence the along-the-channel behaviors, the algorithm reveals the common features of compressible laminar thermal flows. The algorithms do not require the assumption of a linear pressure distribution, which is proved to be untenable in some cases (e.g. constant wall heat flux). The algorithms are validated well by the exact (numerical) computations in exemplary cases of both small and moderate Reynolds number, Mach number and Eckert number of air. Although expressed in a series of eigenfunctions, only several terms (sometimes one or two terms) of solutions are required for a practical computation.

Numerical-analytical method for investigating the stability of motion of bodies of revolution in soft soil media

Abstract A method is presented of investigating the stability of rectilinear motion of a body of revolution in a compressible soil medium with nonlinear physical-mechanical properties of the soil and two-dimensional effects of flow taken into account. The parameters of the axisymmetric process are calculated numerically, whereas the perturbed motion – the radial displacement and rotation relative to the centre of mass – is determined analytically. In the particular case of a conical projectile and linear pressure distribution along the generatrix, an estimate is obtained of the critical position of the centre of mass as a function of the taper angle, the mass and velocity of the body, the coefficient of friction, and the hydrodynamic parameters of the soil medium. Unlike the usually implemented situation of constant pressure postulated by the local interaction models, a displacement of the critical position of the centre of mass by up to 20% of the length of the cone has been found, which leads to a substantial decrease in the margin of stability in a restricted sense. Here, the force parameters and the kinematic parameters of motion of the cone on the boundary of the stability region differ both qualitatively and quantitatively. The stability of motion of bodies in soil media with a nonlinear pressure distribution over the contact surface has not previously been investigated.

Compressible vapour flow in conduits and fractures

We consider the steady flow of a viscous compressible gas through an axisymmetric or two-dimensional porous medium whose properties in the direction of the flow are sufficiently slowly varying. The study is partly motivated by a number of different applications in the Earth sciences, including the release of magmatic volatiles from a magma chamber beneath an active volcano and the discharge of geothermal fluids. The results are also relevant to evaluating the consequences of an accidental release of carbon dioxide from a storage reservoir within the Earth, as might happen at a carbon capture and storage (CCS) site. We consider both slow, thermally equilibrated, flows and fast, adiabatic flows. Because the flow is compressible, it is the mass (and not the volume) flux which is conserved along the flow. We determine this constant mass flux and the velocity and pressure fields, both of which vary with position along the flow, as a function of all the physical parameters. We find that the resultant pressure gradient in the medium is largest at the far, low-pressure end of the conduit because the velocity is largest at that end due to the smallest density being associated with the smallest pressures. This means that the pressure in the permeable conduit is always larger than the linear pressure distribution which joins the given pressures at depth and at the surface, as would be the situation if the flow were incompressible. The detailed pressure distribution is shown to depend on the variation with depth of the quantity $\unicode[STIX]{x1D707}T/(ka^{2})$, where $\unicode[STIX]{x1D707}$ is the dynamic viscosity of the vapour, $T$ is the external temperature, $k$ the permeability and $\unicode[STIX]{x03C0}a^{2}$ the cross-sectional area of the conduit. The resultant mass flux is determined to be proportional to the mean along the flow of $\unicode[STIX]{x1D707}T/(ka^{2})$. We present two numerical illustrations of the results.

A strip anvil apparatus with linear uniform pressure distribution

Bridgman anvil is a useful and effective tool in high-pressure research. However, in this apparatus, the pressure distribution is essentially centrosymmetric. Thus, considerable pressure gradients exist in the gasket and in the sample chamber respectively, and the uniform pressure area is limited. To improve the pressure uniformity in flat face supported anvils, we design a strip face anvil instead of conventional round face anvil and adopt an assortive strip gasket. Principle analysis and a series of pressure calibration experiments are also presented in this paper.The construction of the strip anvil and relevant parts of the apparatus are shown in the diagrams and photos. The relationship between pressure and shearing stress in the strip gasket is investigated by using the model of M. Wakatsuki, which indicates that the pressure distribution should be uniform along the central line of the strip gasket.Pressure calibration experiments are conducted by using strip anvils made of tungsten carbide with a length of 20 mm and width of 5 mm and by using the assortive strip gasket of pyrophyllite. Pressures at different places of the central line are calibrated according to the known phase transitions of bismuth in the same loading process, and the samples are assembled with symmetrical, unsymmerical, and separated local collocations, respectively.Experimental results exhibit that the pressure reaches up to 10 GPa in the central line of the strip gasket, and the pressures are almost equal at least within the range of 12 mm on the central line. The bias errores of oil pressures measured at different places of the central line are all less than 2.0% at 2.55 GPa and 3.6% at 7.7 GPa, indicating only a small pressure gradient along the central line. The main reason for the measuring bias errors lies in the difficulty of the assembly technique. Specifically, the bismuth wire is difficult to adhere to the central line of the anvil during compression. Hence, further improvement of the process is expected in the future.In conclusion, the strip anvil is a unique high-pressure apparatus. The principle analysis and pressure calibration experiments confirm that the pressure is uniform in one-dimensional direction along the central line of the strip anvil. This feature is propitious to the accurate investigation of linear samples under high pressure.

A control volume mixed finite element formulation for tetrahedral elements

SummaryThe control‐volume mixed finite element method is formulated for and applied to a computational domain consisting of a tetrahedral partitioning to solve the steady groundwater flow equations. Test functions consistent with piecewise constant and piecewise linear pressure distributions are used in the formulation. Comparisons are made with a standard mixed finite element formulation using lowest‐order Raviart Thomas basis functions. Results suggest that the control‐volume based formulation is a viable alternative to the standard formulation. Copyright © 2015 John Wiley & Sons, Ltd.

Analytical coupled axial and radial productivity model for steady-state flow in horizontal wells

A new analytical model for coupled radial well inflow and axial flow has been developed and applied to horizontal well productivity calculations. This analytical model results in a linear pressure distribution in the axial direction and a logarithmic pressure distribution in the radial direction. The analytical solution is investigated for two special cases where a one-dimensional analytical solution already exists. It is proven that the model reduces to the classical radial inflow model in the case when the axial pressure gradient is zero and to the linear Darcy equation when there is zero radial inflow. The new equations are also used to evaluate the well flow rates when the skin effect is included. The analytical solution verifies that the axial flow in the reservoir, in general, cannot be ignored, particularly in reservoirs with high permeability and high well productivity. The analytical model is used to quantify the importance of frictional losses in well productivity calculations. An example on how the analytical model can be used in hand calculations to determine well productivity and frictional losses is presented.

The dynamics of a body with an axisymmetric base sliding on a rough plane

In this paper we investigate the dynamics of a body with a flat base sliding on a horizontal and inclined rough plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. For analysis we use the descriptive function method similar to the methods used in the problems of Hamiltonian dynamics with one degree of freedom and allowing a qualitative analysis of the system to be made without explicit integration of equations of motion. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

Experimental study of the oscillation of spheres in an acoustic levitator.

The spontaneous oscillation of solid spheres in a single-axis acoustic levitator is experimentally investigated by using a high speed camera to record the position of the levitated sphere as a function of time. The oscillations in the axial and radial directions are systematically studied by changing the sphere density and the acoustic pressure amplitude. In order to interpret the experimental results, a simple model based on a spring-mass system is applied in the analysis of the sphere oscillatory behavior. This model requires the knowledge of the acoustic pressure distribution, which was obtained numerically by using a linear finite element method (FEM). Additionally, the linear acoustic pressure distribution obtained by FEM was compared with that measured with a laser Doppler vibrometer. The comparison between numerical and experimental pressure distributions shows good agreement for low values of pressure amplitude. When the pressure amplitude is increased, the acoustic pressure distribution becomes nonlinear, producing harmonics of the fundamental frequency. The experimental results of the spheres oscillations for low pressure amplitudes are consistent with the results predicted by the simple model based on a spring-mass system.

Heat transfer coefficient calculated using a linear pressure gradient assumption and measurement for flow boiling in microchannels

Abstract To obtain local flow boiling heat transfer coefficients, some studies used a linear pressure gradient method. In this study, this method is compared with a pressure gradient measurement method. The measurement was carried out for flow boiling of de-ionized water in single horizontal microchannels. Three single microchannels with widths of 0.5 mm, 1 mm and 1.71 mm and a depth of 0.39 mm, giving hydraulic diameters of 0.438 mm, 0.561 mm and 0.635 mm, were tested. The channel length was the same, 62 mm. The nominal mass flux used was 800 kg/m2 s and heat fluxes ranging from 222 to 685 kW/m2 were applied. An inlet fluid temperature of 98 °C and an inlet pressure of 125 kPa (abs) corresponding to an inlet subcooling of 7 K, were maintained at the channel entrance. The trends of heat transfer coefficient decrease with quality at a fixed heat flux and mass flux along the channel. However, the heat transfer mechanism was dominated by nucleate boiling for the 0.561 mm and 0.635 mm channels indicated by the increased local heat transfer coefficient with heat flux, while for the 0.438 mm channel, it was dominated by convective boiling because there was no effect of heat flux on the local heat transfer coefficient. Local heat transfer coefficients determined using the two methods were found to be significantly different, especially at high heat fluxes. When the Φ is less than 9 W/m2 Pa for the 0.438 mm channel, less than 11 W/m2 Pa for the 0.561 mm channel and less than 16 W/m2 Pa for the 0.635 mm channel, the linear pressure distribution may be used with caution.

Vertical stress distribution in isotropic half-spaces due to surface vertical loadings acting over polygonal domains

By integrating the classical Boussinesq expression we derive analytically the vertical stress distribution induced by pressures distributed with arbitrary laws, up to the third order, over polygonal domains. Thus, one can evaluate in closed form either the vertical stress produced by shell elements, modelling raft foundations by finite elements, acting over a Winkler soil or those induced by a linear pressure distribution simulating axial force and biaxial bending moments over a pad foundation. To this end we include charts and tables, both for rectangular and circular domains, which allow the designer to evaluate the vertical stresses induced by linear load distributions by hand calculations. The effectiveness of the proposed approach is witnessed by the comparison between the analytical results obtained with the proposed formulas and the numerical ones of a FEM discretization of the soil associated with the loading distribution induced by a foundation modeled by plate elements resting on a Winkler soil.

Modelling uniformly porous facades to predict dwelling infiltration rates

It is important to limit dwelling infiltration to reduce energy demand and help meet national climate change commitments while concurrently providing sufficient ventilation to deliver adequate indoor air quality. DOMVENT3D is a model of infiltration and exfiltration that assumes a linear pressure distribution over any number of uniformly porous facades and integrates the airflow rate in the vertical plane to predict the theoretically correct airflow rate through them. DOMVENT3D is a new development of an existing two-dimensional model of infiltration that provides more opportunities for investigating a greater number of dwellings than was previously possible. Initial testing suggests that DOMVENT3D is mathematically robust and is suitable for modelling a wide variety of dwelling types and geometries to assist engineers and policy makers. Practical application: The modern building services engineer may be required to model airflow networks in a building to balance the conflicting needs of energy consumption reduction and occupant health. Limiting exfiltration is one method of reducing heat losses from a building and so there is a need to model it accurately. This article presents a new model of infiltration and exfiltration through a uniformly porous facade that can be incorporated within advanced complex airflow network tools or applied using a simple spreadsheet.