Darcy-Forchheimer Equation Research Articles

Evaluation of wave-induced flow through marine porous media accounting for transition of seepage properties across multiple flow regimes

Wave-induced flow through marine porous media has been attracting coastal engineers and researchers because of their strong correlation with scouring, internal erosion, piping and other destructions of marine porous structures. Previous studies chose Darcy-Forchheimer equation for wave-induced seepage behavior analyses because of its simplicity and perceptiveness. However, numerous experiment observations suggested that significant flow regime transition occurred in porous media with high variations in porosity and particle size, while Darcy-Forchheimer equation is only applicable to flow within Forchheimer regime. In this paper, a mathematical model linking the resistance of porous medium to its porosity and particle size is proposed to characterize the seepage properties across Darcy and Forchheimer regime under wave loadings. The proposed seepage model is then incorporated into volume-averaged Reynolds-averaged Navier-Stokes equations, and thus enabling evaluating fluid motions in waves and marine porous media via a unified framework. Through validation against experimental observations, analytical solutions and well accepted computed results in the literature, the proposed model is shown to replicate the characteristics of resistance of porous media with different porosities and particle sizes under different flow regimes. Numerical analyses are conducted to elucidate that seepage velocities of wave-induced flow in marine porous materials can be over- or under-estimated if not considering transitional seepage properties across multiple flow regimes. Such discrepancies can become significant when strong variations occur in porous media with respect to particle size and/or porosity.

Leak-Rate Through Carbon Brush Seals: Experimental Tests Versus Predictions From a Porous Medium Approach

Abstract This study presents a detailed comparative analysis between experimental leakage flow rates and numerical predictions for carbon brush seals with long bristles, utilizing a porous medium model approach. A series of tests were carried out on a static rig (without rotor rotation). The experimental setup allows tests under various interference conditions, revealing significant insights into the flow behavior through the brush seal. A numerical model based on the Darcy–Forchheimer equation is developed to interpret the complex flow dynamics within the brush seal, accounting for viscous, compressible, and inertial effects. The study evaluates the impact of brush deformation and porosity on flow resistance, leveraging experimental data to refine the numerical model parameters. This investigation not only deepens the understanding of brush seal flow physics but also improves the predictive accuracy of the numerical model in simulating operational conditions.

Fin angles optimization of water-cooled plate-fin heat sink based on anisotropic Darcy–Forchheimer theory

Flat heat sinks are devices for efficient heat dissipation and are important components in manufacturing. For example, in precision glass molding, it is utilized as a cooling plate contributing to improve the quality of the product by reducing residual stress and shape deviation, during the gradual cooling phase. In this study, we attempted to achieve precise temperature design of the cooling plate using variable lattice optimization. Fins allow for controlling the flow direction and achieving efficient heat transfer. Therefore, elliptical cylindrical fins were adopted as lattice unit cells, increasing the design freedom of the flow field. To account for the asymmetric flow field within the unit cell, we introduced an asymmetric tensor for the coefficients of the effective physical properties in the Darcy–Forchheimer equation. The proposed method's effectiveness and validity are discussed through numerical calculations with effective material properties, reproduced detailed shapes, and experimental verification. The optimization aiming to minimize the average temperature yielded a numerical simulation temperature decrease of 24.6 % and an experimental temperature decrease of 14.1 % compared to the benchmark model.

Significance of Darcy–forchheimer Casson fluid flow past a non-permeable curved stretching sheet with the impacts of heat and mass transfer

Casson fluid flow based on a curved surface has been mathematically modeled using Brownian motion and the Darcy-Forchheimer equation of porosity. Similarity variables are applied to convert flow-anchored partial differential equations into basic ordinary differential equations. The MATLAB solver “bvp4c” is utilized to derive numerical responses for the problem under consideration. During the numerical approach, all parameters have been set to their values based on reviewed literatures and numerical solutions for all flow fields have been obtained against the concerned parameters. Additionally, a variety of graphs are created utilizing numerical extractions to discuss the results. The curvature parameter (K=1.5,2.5,3.5,4.5) results in an improvement in velocity distribution (f'η=1=0.32564,0.360142,0.378247,0.385401). The reason is that for higher values of the curvature parameter, the radius of a curved sheet decreases. Hence, a smaller region of sheet will be in contact, which produces a small amount of resistance towards fluid particles, so that the velocity profiles show an enhancement. Increasing inputs of Forchheimer number imply stronger inertial effects and it introduce an additional resistance to flow that's why fluid velocity decreased. Additionally, for a curved stretched sheet, a higher drag force is required compared to a flat plate because curved sheet has a larger surface area while flat plate has a minimal surface area in contact with the fluid.

Porous ZrO2–ZrO2 ceramics with excellent mechanical strength and permeability for transpiration cooling

AbstractThe great development of transpiration cooling technology challenges the coolant medium seriously. In this paper, porous ZrO2–ZrO2 ceramics have been satisfactorily prepared to meet the requirements of the coolant medium in transpiration cooling. The results illustrate that porous ZrO2–ZrO2 ceramics have excellent compressive strengths; meanwhile, percentages of compressive strength difference (parallel and perpendicular to the mold‐pressing direction) have an upward trend due to the increasing fiber contents. Besides, the failure mode has a huge distinction between parallelity (45° oblique section) and perpendicularity (wedge shape). These ceramics have unimodal pore size distribution (2–10 µm), and their wettability has a substantial improvement reflected by contact angle from 118° to 0°. Permeability behavior across these ceramics has been accurately described using the Darcy–Forchheimer equation, while obtaining viscous and inertial resistance coefficients, respectively. This work can provide an essential reference for coolant medium in transpiration cooling.

A two-grid characteristic block-centered finite difference method for Darcy-Forchheimer slightly compressible miscible displacement problem

In this paper, a two-grid characteristic block-centered finite difference method is developed for solving a nonlinear Darcy-Forchheimer slightly compressible miscible displacement problem in porous media. The method is proposed for solving the nonlinear system in two steps. In the first step, the nonlinear equations are approximated on a coarse grid using the characteristic block-centered finite difference method. In the second step, the nonlinear system is linearized using Newton's method with a small positive parameter to ensure the differentiability of the nonlinear term |u|u in the Darcy-Forchheimer equation. The proposed two-grid method is rigorously analyzed, in which a priori error estimates of the velocity, pressure, concentration and its flux are provided, and the rates of convergence are O(ε+Δt+H3+h2). Finally, numerical experiments are conducted to demonstrate the effectiveness of the proposed methods by comparing the efficiency, especially in terms of CPU time.

Multi-Phase flow simulation for transient analysis of ballast drainage behavior and shoulder cleaning effect

This paper presents multi-phase flow simulations to analyze the transient transport of rain droplets through rail ballast and its dependence on fouling conditions and shoulder cleaning. The entire process consisting of random rain droplet injection, droplet migration through the air, falling on the ballast surface, drainage through the void spaces between ballast aggregates, and running off the ballasts are modeled. A computational fluid dynamics (CFD) model is developed, in which the volume of fluid (VOF) method and Darcy-Forchheimer's equation are employed to capture the gas–liquid mixture interface and consider fouling-induced flow resistance, respectively. The model is implemented as an interFoam solver in the OpenFOAM CFD tool. A method for determining trapped moisture inside the fouled and moderately fouled ballast layers is also presented. Parametric simulations are conducted to investigate the effect of fouling conditions, more specifically, Fouling Index (FI), Fouling Profile (FP), and shoulder cleaning on ballasts' drainage performance and time-dependent moisture accumulation and penetration. The results indicate that severe congestion in fouled ballast is responsible for higher moisture accumulation, slower penetration, and poorer drainage of ballast. Shoulder cleaning is an effective measure to reopen flow passage and restore drainage capability, in particular, for the fouled ballast.

Permeability measurement of a 3D-printed AlSi10Mg porous medium: comparisons between first results with different experimental and numerical techniques

In this work, the permeability of a 3D-printed, AlSi10Mg porous medium, with porosity ε = 0.3 and an effective pore radius of 48 μm, developed to operate as wick in a sinter-like heat pipe, has been investigated by means of two different experimental approaches, and of two different numerical methods. The two experimental methods are the capillary rise tests, from which permeability was estimated by fitting the theoretical capillary rise curve to the experimental data, and the direct measurement of the the mass flow rate across the porous sample at an imposed pressure difference. The numerical simulations were performed too using two different approaches and software tools, namely, Lattice-Boltzmann with Palabos, and Finite-Volumes with OpenFOAM. In both cases, the simulation domain was reconstructed from a micro-computer aided tomographic scan of a porous medium sample. Preliminary simulations were run on a simple configuration, both to check simulation setup and validate results, and mesh independence was assessed. Then, pressure-driven and velocity-driven simulations of an incompressible fluid flow across the domain were performed, from which the permeability was estimated using Darcy and Darcy-Forchheimer equations. The results show that the methods, while not in complete agreement, provide a useful estimate. The numerical methods also complement the information given by the experimental techniques by highlighting the flow paths, and allow to analyze scenarios of increased and decreased porosity.

Modeling single-phase transverse flows in a PWR rod bundle at low Reynolds number

In normal functioning conditions, the primary component of the flow within the rod bundle of a PWR core is in the axial direction, along the rods. However, in accidental situations, such as the refilling or reflooding phase during a Loss of Coolant Accident, or a Steam Line Break accident, transverse flows may significantly affect thermal-hydraulic properties in the core. Such effects have received little attention in system codes such as CATHARE, particularly in anisotropic rod structures at low Reynolds numbers. Here, we develop macroscopic pressure drop models for flows spanning from the creeping regime to unsteady vortex shedding. The averaged model is a generalized Darcy–Forchheimer equation with an apparent permeability including a rotation matrix and a non-linear dimensionless coefficient. The constitutive relations for the intrinsic permeability matrix are derived from polynomial regressions based on results obtained from microscale numerical simulations at different flow directions and Reynolds numbers. We finally test our approach in a case where transverse flow is imposed through a differential of inlet velocities and validate by comparing results from CATHARE with those obtained from a finite element toolbox.

An improved model for the drying of porous solids with abruptly changing permeability

We present the simulation of the drying of porous solids with abruptly changing permeability using the Darcy-Forchheimer equations. This model arises in fluid mechanics through porous media when the flow exhibits moderate to high velocities and can be useful to predict the drying rate of the first drying period, which is usually constant. The presence of abrupt changes in permeability creates strong singularities and makes it difficult to obtain accurate predictions of the fluid flow, velocity, and pressure drop. In order to overcome these difficulties, we apply an adaptive finite element method based on a simple a posteriori error indicator, reliable and locally efficient, that allows us to refine the mesh over the singularities. We also present a semi-analytical method to obtain an accurate measurement of the apparent permeability of a porous medium. Finally, we provide some numerical experiments that illustrate the performance of the method.

Mechanism of wave-induced flow in reshaping breakwaters

The aim of this work was to investigate the reshaping mechanisms of a reshaping berm breakwater (BB) by assessing the results of a two-dimensional numerical model in OpenFoam. The flow inside and outside the breakwater was numerically simulated. The initial and reshaped form of the breakwater was modelled using the Darcy–Forchheimer equation and k–ϵ closure model. The numerical model was calibrated with and validated against experimental data of wave-induced pressure and water-level fluctuations inside and outside the breakwater. Both the initial and reshaped BB were evaluated for calibration and validation processes. The results show that the minimum run-down level on the breakwater slope is a critical area for armour instability due to outward driving forces caused by the critically synchronised outward flow and excess pressure gradient. Moreover, parallel downward flow occurs during a wave run-down, which can push the displaced armour down the slope. After reshaping, the breakwater profile has a milder slope than the initial profile, and the outward forces due to the excess pressure gradient are reduced to less than one-half of their initial amount, a result of the change in breaker type. Accordingly, the reshaped profile is modified to harmonise with the new flow field conditions.

Flow Characterization in Triply Periodic Minimal Surface (TPMS)-Based Porous Geometries: Part 1—Hydrodynamics

The modeling of flow and heat transfer in porous media systems has always been a challenge, and the extended Darcy transport models are used for macro-level analysis. However, these models are subjected to the limitations depending upon the porous geometry such as pore size, pore type, effective porosity, tortuosity, permeability, and the flow characteristics. The forced convective flow of an incompressible viscous fluid through a channel filled with four different types of porous geometries constructed using the Triply Periodic Minimal Surface (or TPMS) model is presented in this study. Four TPMS lattice shapes, namely Diamond, I-WP, Primitive, and Gyroid, are created with same volume fraction of solid subdomain as 0.68 (or void fraction as 0.32). Using different configurations for the solid subdomain by treating it as (a) solid, (b) fluid, and (c) porous zone, three different classes of porous structures are further generated for each TPMS lattice. The present study is executed with the objective to investigate the effect of shape–morphology, tortuosity, microporosity, and effective porosity on permeability and inertial drag factor. A pore-scale direct numerical simulation approach is performed for the first two types of porous media by solving the Navier–Stokes equations. The specific microporosity is quantitatively induced in the solid subdomain where Darcy–Forchheimer equation is solved, whereas the Navier–Stokes equations are solved for the void subdomain in the third type of porous media. The results reveal that Darcy flow regime exists up to the mean velocity value of U < 0.0025 m/s (Re < 10) for all the cases discussed here, and it deviates at the higher mean velocity. The conductance to the flow shown by Darcy number has the maximum and minimum values for the Primitive Type 2 and I-WP Type 1 cases. The inertial drag coefficient is minimum in Diamond lattice and maximum in Primitive lattice at lower porosity (0.32), while Primitive lattice has minimum and I-WP lattice has maximum value of inertial drag coefficient for higher porosity (~ 1).

Hybrid modelling of saturated water flow in percolating and non-percolating macroporous soil media

The saturated water flow phenomenon is determined by the soil pore transport processes occurring at a micro-scale. In this study, saturated water flow was modeled using two different approaches, depending on the existence of the percolating macropore network. The soil material comprised 26 undisturbed soil cores. Soil samples were scanned using an X-ray micro-CT scanner, and saturated hydraulic conductivity (Ksat), bulk density and particle size distribution were measured. The voxel size for X-ray CT scans of soil cores was 23.6 µm, which allowed soil macropore network discrimination. The macropore network was percolating in 11 samples, while the remaining cores were not. A typical approach based on Navier–Stokes (NS) equations was used for saturated water flow modeling in the case of a percolating samples. In the case of cores with a non-percolating macropore network, the NS modeling approach could not be used. An alternative method of modeling (NS/Darcy) was used in this case, blending: regular NS flow in the well-defined macropores with the Darcy–Forchheimer flow in the remaining part – the soil matrix. Soil matrix is treated by the NS/Darcy model as a pore medium without well-defined pore geometry but with some intrinsic permeability incorporated in the model using the Darcy–Forchheimer equation. Unlike the NS approach, the NS/Darcy model allowed for the simulation of water flow for all soil samples, including those where the macropore network was not percolating. Based on simulations, the Ksat was estimated used for model validation. The analysis of results leads to the proposal of a new hybrid modeling approach, mixing the NS and NS/Darcy modeling approaches. A good estimation of the Ksat was obtained using the proposed model (R2 = 0.61). The NS/Darcy modeling approach was used for the analysis of the macropore flow in the soil media. The simulations show that water permeates through the core, but macropores are a favorable flow path if they exist, even if they are not directly connected to each other. The areas of the soil cores taking part in the preferential, macropore flow were quantified, showing that only a small fraction of the macropores take part in water flow both for percolating and non-percolating cores. But generally, for most of the analyzed flow-related indices, apparent differences in results between percolating and non-percolating samples were observed. Effective flow area (EFA), i.e., the sample area used for water flow with a velocity higher than the threshold velocity (Utr) was analyzed. Considering the macropore flow, only ∼2% of sample volume is responsible for: 82% of the total flux in case of percolated and 34% in case of non-percolated samples. Also, for non-percolated samples, the dependence (R2 = 0.44) between relative flux participation and the effective flow area is observed. The simulation results for the non-percolating samples revealed the relationship between the simulated saturated conductivity of the whole soil sample and the saturated conductivity of the soil matrix and macroporosity. This allowed for developing a simple multiple linear regression model (R2 = 0.98) of the soil core’s hydraulic conductivity.

Implicit Finite-Volume Scheme to Solve Coupled Saint-Venant and Darcy–Forchheimer Equations for Modeling Flow Through Porous Structures

Implicit Finite-Volume Scheme to Solve Coupled Saint-Venant and Darcy–Forchheimer Equations for Modeling Flow Through Porous Structures

Residual-baseda posteriorierror analysis for the coupling of the Navier–Stokes and Darcy–Forchheimer equations

In this paper we consider a mixed variational formulation that have been recently proposed for the coupling of the Navier–Stokes and Darcy–Forchheimer equations, and derive, though in a non-standard sense, a reliable and efficient residual-baseda posteriorierror estimator suitable for an adaptive mesh-refinement method. For the reliability estimate, which holds with respect to the square root of the error estimator, we make use of the inf-sup condition and the strict monotonicity of the operators involved, a suitable Helmholtz decomposition in non-standard Banach spaces in the porous medium, local approximation properties of the Clément interpolant and Raviart–Thomas operator, and a smallness assumption on the data. In turn, inverse inequalities, the localization technique based on triangle-bubble and edge-bubble functions in localLpspaces, are the main tools for developing the efficiency analysis, which is valid for the error estimator itself up to a suitable additional error term. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.

Nonlinear and non-local analytical solution for Darcy–Forchheimer flow through a deformable porous inclusion within a semi-infinite elastic medium

Permeability is a nonlinear and non-local function of the intimate coupling between pore fluid flow and solid deformation in porous media. A class of related problems involves the effect of fluid injection or withdrawal on the transport properties of geomaterials. This paper presents an analytical solution for the nonlinear and non-local problem of fluid flow through a disk-shaped porous elastic inclusion below the free surface of an elastic half-space. The solution accounts for the following nonlinear mechanisms: (i) variations in the permeability coefficient due to the flow-induced deformation of the inclusion; (ii) the inertial losses of the pore fluid flow. The former and latter mechanisms are formulated using the Green's function for a dilatation centre in an elastic half-space and the Darcy–Forchheimer equation for fluid flow through porous media, respectively. An analytical perturbation solution to the considered problem is developed and validated against the numerical finite element solution to the same problem. The described nonlinear mechanisms are represented by two dimensionless parameter groups. The extreme values of these dimensionless groups govern the solution asymptotic behaviours mimicking the special-case solutions in which either mechanism is forced to vanish. The applied aspects of the solution are demonstrated through the wellbore performance index parameter that quantifies the subsurface rock ability to deliver the pore fluid toward or away from a wellbore in a porous reservoir. Unlike the linear models, the presented nonlinear solution captures the observed dependence of the performance index on the wellbore flow rate.

A fully‐mixed finite element method for the coupling of the Navier–Stokes and Darcy–Forchheimer equations

AbstractIn this work we present and analyze a fully‐mixed formulation for the nonlinear model given by the coupling of the Navier–Stokes and Darcy–Forchheimer equations with the Beavers–Joseph–Saffman condition on the interface. Our approach yields non‐Hilbertian normed spaces and a twofold saddle point structure for the corresponding operator equation. Furthermore, since the convective term in the Navier–Stokes equation forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well‐known Schauder and Banach theorems, combined with classical results on nonlinear monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. In particular, given an integer k ≥ 0, Raviart–Thomas spaces of order k, continuous piecewise polynomials of degree ≤k + 1 and piecewise polynomials of degree ≤k are employed in the fluid for approximating the pseudostress tensor, velocity and vorticity, respectively, whereas Raviart–Thomas spaces of order k and piecewise polynomials of degree ≤k for the velocity and pressure, constitute a feasible choice in the porous medium. A priori error estimates and associated rates of convergence are derived, and several numerical examples illustrating the good performance of the method are reported.

Modeling cell venting and gas-phase reactions in 18650 lithium ion batteries during thermal runaway

A numerical model is developed to study cell venting, internal pressure, and gas-phase dynamics behavior of 18650 Li-ion cells undergoing thermal runaway. A k-ϵ Reynolds-Averaged Navier-Stokes (RANS) model is adopted to describe the turbulent flow out of the cells, while the fluid dynamics inside the cells is described by Darcy-Forchheimer's equation. Thermal abuse reactions and gas generation kinetics are described by a single-step lumped reaction model. Then, a series of computational fluid dynamics (CFD) simulations are conducted on a single 18650 cell at various states-of-charge (100%, 50%, 25%) to study detailed flow and thermal behavior as a function of quantity of gas generated during cell venting. Venting events are categorized into two stages: i) breaching of the cell container and ii) thermal runaway reactions. It is found that the cell response is dominated by the second stage since most of the gases are generated during thermal runaway. Also, the propensity for propagation is highly affected by state-of-charge (SOC). Cells at higher SOCs produce more heat and gas during the venting event, owing to higher mass and concentrations of reacting gases, and consequently reach higher internal cell pressures which increase the risk of side-wall breaching.

Thermal storage analysis of a foam-filled PCM heat exchanger subjected to fluctuating flow conditions

A dynamic heat transfer model is developed to investigate the transient thermal storage characteristics of a heat exchanger with foam-filled phase change material (PCM) under the fluctuating flow conditions. The thermal energy storage (TES) configuration of double pipe is considered, while using the highly porous metal foam to intensify the heat transfer in both the PCM and heat transfer fluid (HTF) regions. The paraffin RT50 is used as the PCM and water is selected as the HTF. The energy transport between the inner and outer pipes is thoroughly taken into account. The Darcy-Forchheimer equation is adopted to model the fluid flow through metal foam, and the enthalpy-porosity method is employed to simulate the phase change process within the PCM/foam composite under the local thermal non-equilibrium (LTNE) condition. The performance of heat exchanger during the charging process is firstly predicted under steady HTF flow, and then the effects of fluctuating inlet temperature and velocity are examined respectively. In the case of steady flow, compared with the pure PCM case, inserting metal foam in both the PCM and HTF regions shortens the melting time by 93.6% and augments the heat storage rate by 9 times. Fluctuation in the inlet temperature leads to a visibly oscillating variation in the PCM temperature and heat storage, which are 15 K, 65 kJ at a fluctuation amplitude of 30 K and 17 K, 105 kJ at a periodicity of 400 s. However, fluctuation in the velocity has no considerable influence. The heat exchanger with lower porosity foam presents more sensitive to the fluctuating inlet temperature.

Compound Boussinesq-type modelling over porous beds

In the present work a previously presented high-order Boussinesq-type model is extended to account for wave propagation over porous beds. The resulting wave solver is coupled with a sediment transport and morphological model to form a compound numerical tool for studying coastal hydrodynamics and morphodynamics. The linear properties of the derived equations are studied analytically with respect to wave celerity and spatial damping rate. The values of relevant parameters for the improvement of linear properties are also reviewed. The model is tailored for simulating wave propagation over permeable submerged breakwaters. The effect of the inclusion of dispersive terms in Darcy-Forchheimer equation on the propagation of both short and long waves over a permeable submerged breakwater is studied numerically. A qualitative investigation of the wave-induced current field and the resulting morphological evolution of a sandy beach is also presented.