Small Darcy Number Research Articles

Drag of Rankine bodies embedded in porous media at small Darcy numbers

This paper studies the flow over a solid inclusion embedded in a porous medium of low permeability. We study Rankine bodies which represent circular cylinders and spheres when the aspect ratio is one, and thin plates and almost uniform rods when the aspect ratio is small. Using boundary layer theory, the flow field and pressure distribution are determined. It is found that the pressure drag dominates the shear drag except for small aspect ratios. Thus, geometry plays a decisive role in the choice of using the Darcy equation or the Darcy-Brinkman equation. The results are important for the determination of flow resistance of a non-homogeneous porous medium which has solid inclusions.

Small Darcy number limit of the Navier–Stokes–Darcy system

We study the small Darcy number behavior of the Navier–Stokes–Darcy system with the conservation of mass, Beavers–Joseph–Saffman–Jones condition, and the Lions balance of the normal-force interface boundary conditions imposed on the interface separating the Navier–Stokes flow and Darcy flow. We show that the asymptotic behavior of the coupled system, at small Darcy number, can be captured by two semi-decoupled Darcy number independent sequences: a sequence of (linearized) Navier–Stokes equations, and a sequence of Darcy equations with appropriate initial and boundary data. Approximate solutions to any order of the small parameter (Darcy number) can be constructed via the two sequences. The local in time validity of the asymptotic expansion up to second order is presented. And the global in time convergence is derived under the assumption that the Reynolds number is below a threshold value.

Couette-Poiseuille flow in a fluid overlying an anisotropic porous layer

In this study, Couette-Poiseuille flow inside a channel partially filled with a porous medium is treated. Considering the applicability to the practical scenarios, we assume that the porous medium is anisotropic in nature. More precisely, the model under steady-state assumption is described by the coupling of Brinkman-Forchheimer and Stokes equations. We develop the existence and uniqueness results corresponding to this two-dimensional Stokes-Brinkman-Forchheimer coupled system using the Browder-Minty theorem. For this we restrict the permeability to a diagonal matrix. The coupled system involves a non-linear term due to the Brinkman-Forchheimer equation. We propose an algorithm based on shooting and searching techniques to develop a numerical solution for this system, however, for an arbitrary anisotropic permeability matrix. The robustness of the presented numerical solution is established by comparing it with asymptotic solutions corresponding to small and large Darcy numbers. For the case of a large Darcy number, the problem of interest is a regular perturbation while the same is a singular perturbation problem for the case of a small Darcy number, which is dealt with using Prandtl's matching principle. With the velocity distribution determined, we have presented a comparative analysis of the shear stress distribution at the liquid-porous interface and at the bottom plate, which is of interest to analyze the stress distribution on the arterial wall. The asymptotic results are found to be in good agreement with those obtained numerically.

Predicting convection configurations in coupled fluid–porous systems

A ubiquitous arrangement in nature is a free-flowing fluid coupled to a porous medium, for example a river or lake lying above a porous bed. Depending on the environmental conditions, thermal convection can occur and may be confined to the clear fluid region, forming shallow convection cells, or it can penetrate into the porous medium, forming deep cells. Here, we combine three complementary approaches – linear stability analysis, fully nonlinear numerical simulations and a coarse-grained model – to determine the circumstances that lead to each configuration. The coarse-grained model yields an explicit formula for the transition between deep and shallow convection in the physically relevant limit of small Darcy number. Near the onset of convection, all three of the approaches agree, validating the predictive capability of the explicit formula. The numerical simulations extend these results into the strongly nonlinear regime, revealing novel hybrid configurations in which the flow exhibits a dynamic shift from shallow to deep convection. This hybrid shallow-to-deep convection begins with small, random initial data, progresses through a metastable shallow state and arrives at the preferred steady state of deep convection. We construct a phase diagram that incorporates information from all three approaches and depicts the regions in parameter space that give rise to each convective state.

Darcy\u2013Brinkman\u2013Forchheimer Model for Film Boiling in Porous Media

The paper presents results of the modelling of heat transfer at film boiling of a liquid in a porous medium on a vertical heated wall bordering with the porous medium. Such processes are observed at cooling of high-temperature surfaces of heat pipes, microstructural radiators etc. Heating conditions at the wall were the constant wall temperature or heat flux. The outer boundary of the vapor film was in contact with moving or stationary liquid inside the porous medium. An analytical solution was obtained for the problem of fluid flow and heat transfer using the porous medium model in the Darcy–Brinkman and Darcy–Brinkman–Forchheimer approximation. It was shown that heat transfer at film boiling in a porous medium was less intensive than in the absence of a porous medium (free fluid flow) and further decreased with the decreasing permeability of the porous medium. Significant differences were observed in frames of both models: 20% for small Darcy numbers at Da < 2 for the Darcy–Brinkman model, and 80% for the Darcy–Brinkman–Forchheimer model. In the Darcy–Brinkman model, depending on the interaction conditions at the vapor–liquid interface (no mechanical interaction or stationary fluid), a sharp decrease in heat transfer was observed for the Darcy numbers lower than five. The analytical predictions of heat transfer coefficients qualitatively agreed with the data of Cheng and Verma (Int J Heat Mass Transf 24:1151–1160, 1981) though demonstrated lower values of heat transfer coefficients for the conditions of the constant wall temperature and constant wall heat flux.

Heat transfer analysis of a ventilated room with a porous partition: LB-MRT simulations

This article presents the hydrodynamic and thermal characteristics of transfers by forced, mixed and natural convection in a room ventilated by air displacement. The main objective is to study the effect of a porous partition on the heat transfer and therefore the thermal comfort in the room. The fluid flow future in the cavity and the heat transfer rate on the active wall have been analyzed for different permeabilities: 10−6 ≤ Da ≤ 10. The other control parameters are obviously, the Rayleigh number and the Reynolds number varied in the rows: 10 ≤ Ra ≤ 106 and 50 ≤ Re ≤ 500 respectively. The transfer equations write were solved by the Lattice Boltzmann Multiple Relaxation Time method. For flow in porous media an additional term is added in the standard LB equations, to consider the effect of the porous media, based on the generalized model, the Brinkman-Forchheimer-extended Darcy model. The most important conclusion is that the Darcian regime start for small Darcy number Da &lt; 10−4. Spatial competition between natural convection cell and forced convection movement is observed as Ra and Re rise. The effect of Darcy number values and the height of the porous layer is barely visible with a maximum deviation less than 7% over the ranges considered. Note that the natural convection regime is never reached for low Reynolds numbers. For this Re values the cooperating natural convection only improves transfers by around 10% while, for the other Reynolds numbers the improvement in transfers due to natural and forced convections cooperation is more significant.

Forced convection in a fluid saturated anisotropic porous channel with isoflux boundaries

Fully developed forced convective flow inside a channel filled with a porous material bounded by two impermeable walls subject to a constant heat flux is considered. We consider the Brinkman-Forchheimer equation to govern the flow inside the porous medium, which accounts for the presence of the inertial term. We assume that the porous medium is anisotropic in nature and the permeability is varying along all the directions so that it appears as a positive semidefinite matrix in the momentum equation. We have obtained velocity, temperature, and Nusselt number numerically due to the presence of the nonlinear quadratic term in the momentum equation. Asymptotic solutions for small Darcy number (∼10−3) and high Darcy number (∼10) are obtained. The asymptotic behavior of the Nusselt number is discussed. The key purpose of this paper is to study the effect of anisotropic permeability ratio, anisotropic angle, and inertial parameter on the hydrodynamic quantities and heat transfer for the configuration considered. In particular, we observe that for the moderate range of Darcy number, 10−2 to 102, inertia plays a significant role in the Nusselt number. We observe that inclusion of anisotropic permeability enhances the relative heat transfer rate by almost 20% compared to the corresponding isotropic situation. We present a detailed analysis about the inclusion of the permeability matrix in the Brinkman-Forchheimer extended Darcy momentum equation.

Combined effects of corrugated walls and porous inserts on performance improvement in a heat exchanger channel

In this paper, a numerical study is performed to investigate the combined effects of corrugated walls and porous inserts on heat transfer enhancement and thermal-hydraulic performance in a channel. Accordingly, a sinusoidal wavy wall channel partially filled with a porous material placed at its core is considered. The Darcy-Brinkman-Forchheimehr model is considered to study the flow in porous region and finite volume method (FVM) with SIMPLE algorithm are applied to solve the governing equations. It was found that the amplitude of the wavy wall has a considerable effect on average Nusselt number especially at small Darcy numbers. For example, at Re = 300, δ=0.3, α=0.2 and Da=10−2 the average Nusselt number of wavy channel increases about 12% in comparison with the straight one, while it improves about 38% at Da=10−4. Moreover, there is an optimum value of Darcy number corresponding to maximum performance evaluation criteria (PEC). At δ=0.3 and Re=300, for straight channel and wavy channel with α=0.1 the optimum value of Darcy number occurs at Da=10−3, while for α=0.2 and 0.3 the Da=10−4 leads to the maximum PEC.

Analytical solution of hydrodynamics and heat exchange problem in a porous rectangular channel for thermal boundary conditions of the second kind

In a three-dimensional formulation, an analytical solution of the Darcy-Brinkman equation is obtained together with the two-temperature Schumann model equations for the unidirectional flow of a Newtonian medium in a laminar regime through a horizontal porous channel of a constant rectangular cross-section with known thermal flows at the boundary for small Darcy numbers. The analysis of the specific structure of a compact porous heat exchanger is presented, where the possibilities of the obtained solution for justification its geometrical dimensions during the cooling process of the surface with a heat release of 100 W/cm2 are demonstrated.

On the thermodynamic boundary conditions of a solidifying mushy layer with outflow

AbstractThe free-boundary problem between a liquid region and a mushy layer (a reactive porous medium) must respect both thermodynamic and fluid dynamical considerations. We develop a steady two-dimensional forced-flow configuration to investigate the thermodynamic condition of marginal equilibrium that applies to a solidifying mushy layer with outflow and requires that streamlines are tangent to isotherms at the interface. We show that a ‘two-domain’ approach in which the mushy layer and liquid region are distinct domains is consistent with marginal equilibrium by extending the Stokes equations in a narrow transition region within the mushy layer. We show that the tangential fluid velocity changes rapidly in the transition region to satisfy marginal equilibrium. In convecting mushy layers with liquid channels, a buoyancy gradient can drive this tangential flow. We use asymptotic analysis in the limit of small Darcy number to derive a regime diagram for the existence of steady solutions. Thus we show that marginal equilibrium is a robust boundary condition and can be used without precise knowledge of the fluid flow near the interface.

Numerical simulation on steady flow around and through a porous sphere

The flow around and through a porous sphere is investigated numerically. It is found that the recirculating wake existing downstream of the porous sphere is either completely detached from or penetrates it. Specially, in a certain range of Darcy number, the wake may initially increase in size with an increase in Reynolds number but then decrease in size and eventually disappear when the Reynolds number is sufficiently large. For a small Darcy number, the critical Reynolds number for the onset of recirculating wake may be significantly smaller than that of a solid sphere. The present study suggests that the surface curvature has an important effect on the initial position of the onset of recirculating wake.

Heat transfer from a permeable square cylinder to a flowing fluid

Flow and heat transfer from an isolated square cylinder maintained at a constant temperature is considered. The cylinder is porous with constant permeability and porosity. The cylinder is placed horizontally and is subjected to an uniform flow of air. Forced convection heat transfer from the cylinder is analyzed for different values of Darcy numbers (10 −6–10 −2) at low Reynolds numbers. Brinkman model with an additional Forchheimer term has been used to model the flow in the porous media. The effect of Darcy number and porosity on the drag coefficient, length of the wake behind the cylinder, Nusselt number at each face of the cylinder and the overall Nusselt number of the cylinder have been analyzed for Reynolds numbers 1–40. The important results of the study can be summarised as follows: • The flow is steady for the range of Reynolds and Darcy numbers considered here. The flow pattern through and around the porous cylinder depends much on the Darcy number of the porous medium. At small Darcy numbers (=10 −6), the permeable cylinder behaves like a solid cylinder with no fluid penetrating the surface. With increasing Darcy numbers the fluid starts penetrating the surface with ease. • The drag coefficient, mean Nusselt number and wake length of the permeable square cylinder approaches the corresponding case of a impermeable cylinder placed in an unbounded domain for very low Darcy number, typically at Darcy numbers 10 −6. With increasing Darcy number the drag coefficient and wake length decreases. Flow separation from the cylinder is delayed with increasing Darcy number. • Fluid impinging the front surface carries away more heat than any other surface resulting in a high local Nusselt number along the front surface which increases with increase in Darcy number while it decreases along the other three faces. But the mean Nusselt number of the cylinder shows an enhancement in heat transfer rates with increasing Darcy numbers and Reynolds number.

THE ROLE OF POROUS MEDIA IN MODELING FLUID FLOW WITHIN HOLLOW FIBER MEMBRANES OF THE TOTAL ARTIFICIAL LUNG

A numerical study was conducted to analyze fluid flow within hollow fiber membranes of the artificial lungs. The hollow fiber bundle was approximated using a porous media model. In addition, the transport equations were solved using the finite-element formulation based on the Galerkin method of weighted residuals. Comparisons with previously published work on the basis of special cases were performed and found to be in excellent agreement. A Newtonian viscous fluid model for the blood was used. Different flow models for porous media, such as the Brinkman-extended Darcy model, Darcy's law model, and the generalized flow model, were considered. Results were obtained in terms of streamlines, velocity vectors, and pressure distribution for various Reynolds numbers and Darcy numbers. The results from this investigation showed that the pressure drop across the artificial lung device increased with an increase in the Reynolds number. In addition, the pressure drop was found to increase significantly for small Darcy numbers.

Visualization of heat transport due to natural convection for hot materials confined within two entrapped porous triangular cavities via heatline concept

This paper analyzes the detailed heat transfer within two entrapped porous triangular cavities involving cold inclined walls and hot horizontal walls. A penalty finite element analysis with bi-quadratic elements is performed to investigate the results in terms of isotherms, streamlines and heatlines and local and average Nusselt numbers. The parameters for this study are Darcy number, Da ( 10 - 5 – 10 - 3 ) , Prandtl number, Pr ( 0.015 – 1000 ) and Rayleigh number ( 10 3 – 5 × 10 5 ) . It has been found that at small Darcy number ( Da = 10 - 5 ) , heat transfer is primarily conduction dominant and heatlines are found to be orthogonal to the isotherms. The presence of multiple circulations in streamlines and heatlines are observed within the lower triangle at small Prandtl number Pr = 0.015 with high Darcy number ( Da = 10 - 3 ) whereas only single pair of circulations are observed for higher Prandtl numbers. The convective cells in heatlines gradually become enhanced as Pr increases from 0.015 to 1000. In contrast, variation of Prandtl number gives negligible change in heating pattern within the upper triangle and intensity of streamlines and heatlines are less irrespective of Prandtl number. Heat transfer rates are estimated in terms of local ( Nu l , Nu h ) and average Nusselt numbers ( Nu l ¯ , Nu h ¯ ) . Heat transfer rates are also explained based on heatlines. Local Nusselt numbers with spatial distribution exhibit monotonic trend irrespective of Da and Pr for the upper triangle whereas wavy distribution of local Nusselt number occur for the lower triangle. For Da = 10 - 3 , average Nusselt numbers ( Nu h ¯ and Nu l ¯ ) increase exponentially with Ra at higher Rayleigh numbers. But, overall lower heat transfer rates are observed for the upper triangle. Finally, it is concluded that lower triangle has always has higher heat recovery capacity compared to upper triangle. To achieve efficient heat transfer, fluids with high Prandtl numbers are recommended for the lower triangle whereas any fluid with any Prandtl number may be acceptable for the upper triangle.

Natural convection and flow simulation in differentially heated isosceles triangular enclosures filled with porous medium

A finite element analysis is performed to investigate the effects of uniform and non-uniform heating of bottom wall on natural convection flows within isosceles triangular enclosures filled with porous medium. The detailed analysis is carried out in two cases depending on various thermal boundary conditions: (I) two inclined walls are maintained at constant cold temperature while the bottom wall is uniformly heated; (II) two inclined walls are maintained at constant cold temperature while the bottom wall is non-uniformly heated. The present numerical procedure adopted in this investigation yields consistent performance over a wide range of parameters of Darcy number, Da ( 10 - 5 ⩽ Da ⩽ 10 - 3 ), Rayleigh number, Ra ( 10 3 ⩽ Ra ⩽ 10 6 ) and Prandtl number, Pr ( 0.026 ⩽ Pr ⩽ 1000 ) in all the cases mentioned above. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. It is observed that at small Darcy numbers, the heat transfer is primarily due to conduction irrespective of Pr. As the Darcy number increases, there is a change from conduction dominant regime to convection dominant regime. Flow circulations are also found to be strong functions of Pr at large Da ( Da = 10 - 3 ) and multiple circulation cells occur at small Pr with Ra = 10 6 . Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case, but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. As average Nusselt number is same on both the inclined walls, the average Nusselt number for bottom wall is 2 times that of the inclined wall which is well matched in two cases considered for verifying the thermal equilibrium of the system. The correlations are proposed for average Nusselt number as functions of Ra for various Darcy and Prandtl numbers.

A theoretical analysis of forced convection in a porous-saturated circular tube: Brinkman–Forchheimer model

Fully developed forced convection inside a circular tube filled with saturated porous medium and with uniform heat flux at the wall is investigated on the basis of a Brinkman-Forchheimer model. The matched asymptotic expansion method is applied at small Darcy numbers. For large Darcy numbers, the solution for the Brinkman-Forchheimer momentum equation is found in terms of an asymptotic expansion. Once the velocity distribution is determined, the energy equation is solved using the same asymptotic technique. The results for the two limiting cases of clear fluid and Darcy flow conditions show good agreement with those available in the literature.

Non-Darcy Couette Flow in a Porous Medium Filled With an Inelastic Non-Newtonian Fluid

Newtonian and non-Newtonian Couette flows through inelastic fluid saturated porous media due to a moving plate boundary have been investigated analytically. The momentum equation which includes both the viscous and inertia terms is solved to examine the effects of the pseudoplasticity, boundary friction, and porous inertia on the velocity profile and wall shear stress at the moving wall. Closed-form exact solutions are presented for three distinct cases of practical interest. A simple expression for the wall shear stress valid for all Newtonian and non-Newtonian cases of small Darcy number has been also derived through an integral treatment, and found to agree closely with the exact values obtained from direct numerical integrations of the momentum equation. The analytical results presented here can be used for possible applications to highly sensitive viscosity measuring devices.

Theoretical and experimental heat and mass transfer in highly porous media

The heat and mass transfer coefficients around the porous medium put on the flat plate at a distance from the leading edge of the flat plate are calculated numerically for two-dimensional laminar flows. To verify the analytical model developed and invoke the heat/mass transfer analogy, an experiment is carried out by the naphthalene sublimation technique for the case of negligible radiation field. From the effects of the wake, the Sherwood number is maximum around the region where the porous medium is attached. The theoretical results correspond well with the experimental results at small Darcy number. Permeability of ceramic blocks used for experiment is also measured and the Forchheimer equation is applicable in our measurement range.