Density Maximum Effect on Double-diffusive Natural Convection in a Porous Cavity with Variable Wall Temperature

The effect of density maximum of water on double-diffusive natural convection in a two-dimensioned cavity filled with a water saturated isotropic porous medium is studied numerically. The horizontal walls of the cavity are insulated. The opposing vertical walls are kept at different temperatures θ _h (linearly varies with height) and θ _c (θ _c ≤ θ _h ). The concentration levels at cold wall and hot wall are, respectively, c ₁ and c ₂ with c ₁ > c ₂. Brinkman-Forchheimer extended Darcy model is used to investigate the average heat and mass transfer rates. The non-dimensional equations for momentum, energy, and concentration are solved by finite volume method with power law scheme for convection and diffusion terms. The results are presented in the form of streamlines, isotherms, and isoconcentration lines for various values of Grashof numbers, Schmidt number, porosity, and Darcy numbers. It is observed that the density maximum of water has profound effect on the thermosolutal convection. The effects of different parameters on the velocity, temperature, and species concentrations are also shown graphically.     

Relative permeability and capillary pressure functions of porous media as related to the displacement growth pattern

Visualization experiments of the unsteady immiscible displacement of a fluid by another are performed on glass-etched pore networks of well-controlled morphology by varying the fluid system and flow conditions. The measured transient responses of the fluid saturation and pressure drop across the porous medium are introduced into numerical solvers of the macroscopic two-phase flow equations to estimate the non-wetting phase, k_rnw, and wetting phase, k_rw, relative permeability curves and capillary pressure, P_c, curve. The correlation of k_rnw, k_rw, and P_c with the displacement growth pattern is investigated. Except for the capillary number, wettability, and viscosity ratio, the immiscible displacement growth pattern in a porous medium may be governed by the shear-thinning rheology of the injected or displaced fluid, and the porous sample length as compared to the thickness of the frontal region. The imbibition k_rnw increases as the flow pattern changes from compact displacement to viscous fingering or from viscous to capillary fingering. The imbibition k_rw increases as the flow pattern changes from compact displacement or capillary fingering to viscous fingering. As the shear-thinning behaviour of the NWP strengthens and/or the contact angle decreases, then the flow pattern is gradually dominated by irregular interfacial configurations, and the imbibition k_rnw increases. The imbibition P_c is a decreasing function of the capillary number or increasing function of the injected phase viscosity in agreement with the linear thermodynamic theory.     

On the Chebyshev collocation spectral approach to stability of fluid flow in a porous medium

In this paper, the temporal development of small disturbances in a pressure‐driven fluid flow through a channel filled with a saturated porous medium is investigated. The Brinkman flow model is employed in order to obtain the basic flow velocity distribution. Under normal mode assumption, the linearized governing equations for disturbances yield a fourth‐order eigenvalue problem, which reduces to the well‐known Orr–Sommerfeld equation in some limiting cases solved numerically by a spectral collocation technique with expansions in Chebyshev polynomials. The critical Reynolds number Re_c, the critical wave number α_c, and the critical wave speed c_c are obtained for a wide range of the porous medium shape factor parameter S. It is found that a decrease in porous medium permeability has a stabilizing effect on the fluid flow. Copyright © 2008 John Wiley & Sons, Ltd.     

On Temporal Stability Analysis for Hydromagnetic Flow in a Channel Filled with a Saturated Porous Medium

In this paper, a linear stability analysis is presented to trace the time evolution of an infinitesimal, two-dimensional disturbance imposed on the base flow of an electrically conducting fluid in a channel filled with a saturated porous medium under the influence of a transversely imposed magnetic field. An eigenvalue problem is obtained and solved numerically using the Chebyshev collocation spectral method. The critical Reynolds number Re _c, the critical wave number α _c and the critical wave speed c _c are obtained for a wide range of the porous medium shape factor parameter S and Hartmann number H. It is found that an increase in the magnetic field intensity and a decrease in porous medium permeability have a stabilizing effect on the fluid flow.     

Effect of Wetting on Gravity Drainage in Porous Media

A series of benchmark experiments on the effect of the wetting state on the flow properties in porous media were performed, allowing us to relate the wetting properties at the pore scale to the macroscale hydrodynamics. Drainage of n-alkanes (oils) displaced by air in a model porous medium consisting of water-wet sand was studied using gamma-ray densitometry and weight measurements. The enormous advantage of our system is that we know and control the wetting properties perfectly: we can tune the wetting properties by changing the salinity of the water. This allows us to perform porous medium flow experiments for the different wetting states without changing the transport properties (viscosity and density) of the oil. Drainage is found to be more efficient, and consequently oil recovery more important for partial wetting.     

Unsteady Natural Convection Flow in a Square Cavity Filled with a Porous Medium Due to Impulsive Change in Wall Temperature

Unsteady natural convection flow in a two-dimensional square cavity filled with a porous material has been studied. The flow is initially steady where the left-hand vertical wall has temperature T _h and the right-hand vertical wall is maintained at temperature T _c (T _h > T _c) and the horizontal walls are insulated. At time t > 0, the left-hand vertical wall temperature is suddenly raised to which introduces unsteadiness in the flow field. The partial differential equations governing the unsteady natural convection flow have been solved numerically using a finite control volume method. The computation has been carried out until the final steady state is reached. It is found that the average Nusselt number attains a minimum during the transient period and that the time required to reach the final steady state is longer for low Rayleigh number and shorter for high Rayleigh number.     

Double diffusive natural convection in a partially heated enclosure with Soret and Dufour effects

The effect of double-diffusive natural convection of water in a partially heated enclosure with Soret and Dufour coefficients around the density maximum is studied numerically. The right vertical wall has constant temperature θ_c, while left vertical wall is partially heated θ_h, with θ_h > θ_c. The concentration in right wall is maintained higher than left wall (C_c < C_h) for case I, and concentration is lower in right wall than left wall (C_h > C_c) for case II. The remaining left vertical wall and the two horizontal walls are considered adiabatic. Water is considered as the working fluid. The governing equations are solved by control volume method using SIMPLE algorithm with QUICK scheme. The effect of the various parameters (thermal Rayleigh number, center of the heating location, density inversion parameter, Buoyancy ratio number, Schmidt number, and Soret and Dufour coefficients) on the flow pattern and heat and mass transfer has been depicted. Comprehensive Nusselt and Sherwood numbers data are presented as functions of the governing parameters mentioned above.     

Numerical study of Prandtl effect on MHD flow at a lid‐driven porous cavity

In this paper, the lattice Boltzmann method is used to study the Prandtl number effect on flow structure and heat transfer rates in a magnetohydrodynamic flow mixed convection in a lid‐driven cavity filled with a porous medium. The right and left walls are at constant but different temperatures (θ_h and θ_c), while the other walls are adiabatic. Gallium and salt water (0.02 < Pr < 13.4) are used as samples of the electroconducting fluids in the cavity. Typical sets of streamlines and isotherms are presented to analyze the flow patterns set up by the competition among the forced flow created by the lid‐driven wall, the buoyancy force of the fluid and the magnetic force of the applied magnetic field. Mathematical formulations in the porous media were constructed based on the Brinkman–Forchheimer model, while the multidistribution‐function model was used for the magnetic field effect. Numerical results were obtained and the effects of the Prandtl number and the other effective parameters such as Richardson, Hartman, and Darcy numbers were investigated. It was found that the fluid fluctuations within the cavity were reduced by increasing the Hartman number. A similar pattern was observed for the Darcy number reduction. Heat transfer was essentially dominated by the conduction for the low Prandtl number and forced convection dominated as the Prandtl number increased. Also, the average Nusselt number was raised by increasing the Prandtl number. It was discovered that a remarkable heat transfer enhancement of up to 28% could be reached by increasing the Prandtl number (from 0.02 to 13.4) at constant Richardson and Darcy numbers. Copyright © 2011 John Wiley & Sons, Ltd.     

Capillary hyperdisperson of wetting liquids in fractal porous media

Recent displacement experiments show anomalously rapid spreading of water during imbibition into a prewet porous medium. We explain this phenomenon, calledhyperdispersion, as viscous flow along fractal pore walls in thin films of thicknessh governed by disjoining forces and capillarity. At high capillary pressure, total wetting phase saturation is the sum of thin-film and pendular stucture inventories:S _w =S _tf +S _ps . In many cases, disjoining pressure is inversely proportional to a powerm of film thicknessh, i.e. h _–m , so thatS _tf P _c ^–1/m. The contribution of fractal pendular structures to wetting phase saturation often obeys a power lawS _ps P _c ^(3–D), whereD is the Hausdorff or fractal dimension of pore wall roughness. Hence, if wetting phase inventory is primarily pendular structures, and if thin films control the hydraulic resistance of wetting phase, the capillary dispersion coefficient obeysD _c S _w ^v , where v=[3–m(4–D) ]/m(3–D). The spreading ishyperdispersive, i.e.D _c (S _w ) rises as wetting phase saturation approaches zero, ifm>3/(4–D),hypodispersive, i.e.D _c (S ₂) falls as wetting phase saturation tends to zero, ifm<3>D), anddiffusion-like ifm=3/(4–D). Asymptotic analysis of the capillary diffusion equation is presented.     

A Discussion of the Effect of Tortuosity on the Capillary Imbibition in Porous Media

In the past decades, there was considerable controversy over the Lucas–Washburn (LW) equation widely applied in capillary imbibition kinetics. Many experimental results showed that the time exponent of the LW equation is less than 0.5. Based on the tortuous capillary model and fractal geometry, the effect of tortuosity on the capillary imbibition in wetting porous media is discussed in this article. The average height growth of wetting liquid in porous media driven by capillary force following the [`(L)] _s(t) ~ t^1/2D_T{\overline L _{\rm {s}}(t)\sim t^{1/{2D_{\rm {T}}}}} law is obtained (here D _T is the fractal dimension for tortuosity, which represents the heterogeneity of flow in porous media). The LW law turns out to be the special case when the straight capillary tube (D _T = 1) is assumed. The predictions by the present model for the time exponent for capillary imbibition in porous media are compared with available experimental data, and the present model can reproduce approximately the global trend of variation of the time exponent with porosity changing.     

On the Convection in a Porous Medium with Inclined Temperature Gradient and Vertical Throughflow. Part I. Normal Modes

In this second part of our analysis of the destabilization of transverse modes in an extended horizontal layer of a saturated porous medium with inclined temperature gradient and vertical throughflow, we apply the mathematical formalism of absolute and convective instabilities to studying the nature of the transition to instability of such modes by assuming on physical grounds that the transition is triggered by growing localized wavepackets. It is revealed that in most of the parameter cases treated in the first part of the analysis (Brevdo and Ruderman 2009), at the transition point the evolving instability is convective. Only in the cases of zero horizontal thermal gradient, and in the cases of zero vertical throughflow and the horizontal Rayleigh number R _h < 49, the instability is absolute implying that, as the vertical Rayleigh number, R _v, increases passing through its critical value, R _vc, the destabilization tends to affect the base state throughout and eventually destroys it at every point in space. For the parameter values considered, for which the destabilization has the nature of convective instability, we found that, as R _v, increases beyond the critical value, while the horizontal Rayleigh number, R _h, and the Péclet number, Q _v, are kept fixed, the flow experiences a transition from convective to absolute instability. The values of the vertical Rayleigh number, R _v, at the transition from convective to absolute instability are computed. For convectively unstable, but absolutely stable cases, the spatially amplifying responses to localized oscillatory perturbations, i.e., signaling, are treated and it is found that the amplification is always in the direction of the applied horizontal thermal gradient.     

Structuration de la convection mixte en milieu poreux confiné latéralement et chauffé par le bas : effets d''inertiePattern formation of mixed convection in a porous medium confined laterally and heated from below: effect of inertia

We consider the onset of convection in a porous medium heated from below and subjected to a horizontal mean flow. The effect of porous inertia is studied, and the transverse aspect ratio a of the medium is taken into accout. We find that the dominant modes are longitudinal rolls (L.R) if a is an integer or transverse traveling rolls (T.R) if a is below a_c with a_c<1. When a is not an integer with a>a_c, the setting on patterns are oscillatory three-dimensional structures (3D) for a>1 or T.R for a_c<a<1 provided that the Reynolds number remains below a critical value Re_K^*. We show that these structures are replaced by L.R if Re_K>Re_K^*. To cite this article: A. Delache et al., C. R. Mecanique 330 (2002) 885–891.     

Condensation and isothermal water transfer in cement mortar: Part II - transient condensation of water vapor

The transient wetting of a mortar sample swept by a flow of humid air is experimentally studied at temperatures of 30 and 55°C. The water content profile shape and evolution are found to be very different from those which were observed during imbibition. The boundary condition on the exposed wall of the sample is examined. A convenient evolution of the coefficient of diffusion with water content is explored. This coefficient is interpreted in terms of pure vapor diffusion, even at relatively high water contents. But its values at low water content and its temperature dependence are inconsistent. Additional explanations are then considered with the assumption that the vapor condensation in the medium is not an equilibrium process between vapor and liquid phases. The physical origin of such a nonequilibrium process is discussed. A tentative set of transfer and phase change coefficients is proposed in order to describe the experimental data by means of numerical simulation. Then, some aspects of the imbibition processes are re-examined, taking into account the consequences of a nonequilibrium condensation.Nomenclature volumic rate of phase change - D ₀ coefficient of free diffusion of the water vapor in air - D _hv vapor diffusion coefficient of the medium - E, E equivalent air thickness - h relative humidity of gaseous phase - h _c relative humidity at the capillary condensation threshold - h _a relative humidity of the flowing air - h ₀ relative humidity at the air-material interface - h _E equilibrium relative humidity at a given water content - J global massic flux - M molar mass of water - R gas constant - T temperature - t time - x distance from the interface - ₀ total porosity - volumetric water content - _h condensation coefficient (see Equation (8)) - _L mass density of liquid water - vs mass density of saturated water vapor     

Mixed convection in the presence of horizontal impermeable surfaces in saturated porous media with variable permeability

Boundary layer approximation is applied for mixed convection about a horizontal flat plate in a saturated porous medium with aiding external flows. Similarity solutions are obtained, incorporating the variation of permeabilty, for 1) horizontal flat plate at zero angle of attack with constant heat flux; 2) stagnation point flows about horizontal flat plates with wall temperature varying asT _w ∝x ². The temperature and velocity profiles for different values of Ra/(RePr)^3/2 and the parameters governing the flow are obtained. The heat transfer rate is calculated and its implications in a geothermal application is discussed. Further, the criteria for pure mixed convection about horizontal flat plates in a porous media are established.     

Mixed Convection in Water Near the Density Extremum Along a Vertical Plate with Sinusoidal Surface Temperature Variation Embedded in a Porous Medium

A steady laminar boundary layer flowing along a vertical plate immersed in a Darcy–Brinkman porous medium saturated with water at 4°C is studied. The plate temperature varies sinusoidally along the plate between 0 and 8°C where the density of water varies parabolically and is almost symmetrical at about 4°C. Except for the existence of the buoyancy force, it is assumed that either the plate moves upwards or the ambient water moves upwards (moving stream). The results are obtained with the direct numerical solution of the boundary layer equations taking into account the temperature dependence of water thermophysical properties (ρ, μ and c _p). Results are presented for the wall temperature gradient and the wall shear stress along the plate for free convection and mixed convection. Temperature and velocity profiles are also presented.     

Two-Phase Flow in Porous Media: Predicting Its Dependence on Capillary Number and Viscosity Ratio

Motivated by the need to determine the dependencies of two-phase flow in a wide range of applications from carbon dioxide sequestration to enhanced oil recovery, we have developed a standard two-dimensional, pore-level model of immiscible drainage, incorporating viscous and capillary effects. This model has been validated through comparison with several experiments. For a range of stable viscosity ratios (M = μ _injected,nwf/μ _{defending, wf} ≥ 1), we had increased the capillary number, N _c and studied the way in which the flows deviate from fractal capillary fingering at a characteristic time and become compact for realistic capillary numbers. This crossover has enabled predictions for the dependence of the flow behavior upon capillary number and viscosity ratio. Our results for the crossover agreed with earlier theoretical predictions, including the universality of the leading power-law indicating its independence of details of the porous medium structure. In this article, we have observed a similar crossover from initial fractal viscous fingering (FVF) to compact flow, for large capillary numbers and unstable viscosity ratios M < 1. In this case, we increased the viscosity ratio from infinitesimal values, and studied the way in which the flows deviate from FVF at a characteristic time and become compact for non-zero viscosity ratios. This crossover has been studied using both our pore-level model and micro-fluidic flow-cell experiments. The same characteristic time, τ = 1/M ^0.7, satisfactorily describes both the pore-level results for a range of large capillary numbers and the micro-fluidic flow cell results. This crossover should lead to predictions similar to those mentioned above.     

Natural Convection in a Cavity Filled with Porous Layers on the Top and Bottom Walls

Natural convection in a partially filled porous square cavity is numerically investigated using SIMPLEC method. The Brinkman-Forchheimer extended model was used to govern the flow in the porous medium region. At the porous-fluid interface, the flow boundary condition imposed is a shear stress jump, which includes both the viscous and inertial effects, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The results are presented with flow configurations and isotherms, local and average Nusselt number along the cold wall for different Darcy numbers from 10⁻¹ to 10⁻⁶, porosity values from 0.2 to 0.8, Rayleigh numbers from 10³ to 10⁷, and the ratio of porous layer thickness to cavity height from 0 to 0.50. The flow pattern inside the cavity is affected with these parameters and hence the local and global heat transfer. A modified Darcy–Rayleigh number is proposed for the heat convection intensity in porous/fluid filled domains. When its value is less than unit, global heat transfer keeps unchanged. The interfacial stress jump coefficients β ₁ and β ₂ were varied from  −1 to +1, and their effects on the local and average Nusselt numbers, velocity and temperature profiles in the mid-width of the cavity are investigated.     

The structure of the turbulent boundary layer over a mobile and deformable boundary

The structure of the velocity field above a propagating water wave of fixed frequency was investigated in order to evaluate the transport of wind momentum to water waves and the influence of a mobile and deformable boundary on the bursting cycle. The vertical and horizontal velocities were measured in a transformed Eulerian wave-following frame of reference with the aid of a cross hot film, in a wind-wave research facility at Stanford University.The mean velocity profiles have a log-linear form with a wake free-stream characteristic. The wave-coherent motion in the free-stream is irrotational; in the boundary layer, it has a strong shear behavior related to the wave-associated stress. The wave-induced velocity field and the wave-perturbed turbulence depend strongly on the ratio of the wave-speed to the mean free-stream velocity, c/U ₀.The presence of the propagating waves affects the bursting cycle, making the contribution of sweeps and ejections almost equal and dependent on the ratio c/U ₀. The magnitudes of the contribution of the bursting events are generally enhanced by the presence of water waves. The time interval between ejections or sweeps does not scale with either the inner and/or outer flow variables.This paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

Turbulence in rough-wall boundary layers: universality issues

Wind tunnel measurements of turbulent boundary layers over three-dimensional rough surfaces have been carried out to determine the critical roughness height beyond which the roughness affects the turbulence characteristics of the entire boundary layer. Experiments were performed on three types of surfaces, consisting of an urban type surface with square random height elements, a diamond-pattern wire mesh and a sand-paper type grit. The measurements were carried out over a momentum thickness Reynolds number (Re _θ) range of 1,300–28,000 using two-component Laser Doppler anemometry (LDA) and hot-wire anemometry (HWA). A wide range of the ratio of roughness element height h to boundary layer thickness δ was covered (0.04 ￡ h/d ￡ 0.400.04 \leq h/\delta \leq 0.40). The results confirm that the mean profiles for all the surfaces collapse well in velocity defect form up to surprisingly large values of h/δ, perhaps as large as 0.2, but with a somewhat larger outer layer wake strength than for smooth-wall flows, as previously found. At lower h/δ, at least up to 0.15, the Reynolds stresses for all surfaces show good agreement throughout the boundary layer, collapsing with smooth-wall results outside the near-wall region. With increasing h/δ, however, the turbulence above the near-wall region is gradually modified until the entire flow is affected. Quadrant analysis confirms that changes in the rough-wall boundary layers certainly exist but are confined to the near-wall region at low h/δ; for h/δ beyond about 0.2 the quadrant events show that the structural changes extend throughout much of the boundary layer. Taken together, the data suggest that above h/δ ≈ 0.15, the details of the roughness have a weak effect on how quickly (with rising h/δ) the turbulence structure in the outer flow ceases to conform to the classical boundary layer behaviour. The present results provide support for Townsend’s wall similarity hypothesis at low h/δ and also suggest that a single critical roughness height beyond which it fails does not exist. For fully rough flows, the data also confirm that mean flow and turbulence quantities are essentially independent of Re _θ; all the Reynolds stresses match those of smooth-wall flows at very high Re _θ. Nonetheless, there is a noticeable increase in stress contributions from strong sweep events in the near-wall region, even at quite low h/δ.