Finite Element Modelling of Flow Through a Porous Medium Between Two Parallel Plates Using The Brinkman Equation

Despite the widespread use of the Darcy equation to model porous flow, it is well known that this equation is inconsistent with commonly prescribed no slip conditions at flow domain walls or interfaces between different sections. Therefore, in cases where the wall effects on the flow regime are expected to be significant, the Darcy equation which is only consistent with perfect slip at solid boundaries, cannot predict velocity and pressure profiles properly and alternative models such as the Brinkman equation need to be considered. This paper is devoted to the study of the flow of a Newtonian fluid in a porous medium between two impermeable parallel walls at different Darcy parameters (Da). The flow regime is considered to be isothermal and steady. Three different flow regimes can be considered using the Brinkman equation: free flow (Da > 1), porous flow (high permeability, 1 > Da > 10⁻⁶) and porous flow (low permeability Da < 10⁻⁶). In the present work the described bench mark problem is used to study the effects of solid walls for a range of low to high Darcy parameters. Both no-slip and slip conditions are considered and the results of these two cases are compared. The range of the applicability of the Brinkman equation and simulated results for different cases are shown.     

Numerical Study of Natural Convection Heat Transfer in an Inclined Porous Cavity with Time-Periodic Boundary Conditions

Kalabin et al. (Numer. Heat Transfer A 47, 621-631, 2005) studied the unsteady natural convection for the sinusoidal oscillating wall temperature on one side wall and constant average temperature on the opposing side wall. The present article is on the unsteady natural convective heat transfer in an inclined porous cavity with similar temperature boundary conditions as those of Kalabin et al. The inclined angle of the cavity is varied from 0° to 80°. The flow field is modeled with the Brinkman-extended Darcy model. The combined effects of inclination angle of the enclosure and oscillation frequency of wall temperature are studied for Ra* = 10³, Da = 10⁻³, , and Pr=1. Some results are also obtained with the Darcy–Brinkman–Forchheimer model and Darcy’s law and are compared with the present Brinkman-extended Darcy model. The maximal heat transfer rate is attained at the oscillating frequency f = 46.7π and the inclined angle .     

Non-Darcy buoyancy driven flows in a fluid saturated porous medium: the use of asymptotic computational fluid dynamics (ACFD) approach

Natural convection in a fluid saturated porous medium has been numerically investigated using a generalized non-Darcy approach. The governing equations are solved by using Finite Volume approach. First order upwind scheme is employed for convective formulation and SIMPLE algorithm for pressure velocity coupling. Numerical results are presented to study the influence of parameters such as Rayleigh number (10⁶ ≤Ra ≤10⁸), Darcy number (10⁻⁵ ≤ Da ≤ 10⁻²), porosity (0.4 ≤ ɛ ≤ 0.9) and Prandtl number (0.01 ≤ Pr ≤ 10) on the flow behavior and heat transfer. By combining the method of matched asymptotic expansions with computational fluid dynamics (CFD), so called asymptotic computational fluid dynamics (ACFD) technique has been employed to generate correlation for average Nusselt number. The technique is found to be an attractive option for generating correlation and also in the analysis of natural convection in porous medium over a fairly wide range of parameters with fewer simulations for numerical solutions.     

Convection in a Lid-Driven Heat-Generating Porous Cavity with Alternative Thermal Boundary Conditions

Mixed convection flow in a two-sided lid-driven cavity filled with heat-generating porous medium is numerically investigated. The top and bottom walls are moving in opposite directions at different temperatures, while the side vertical walls are considered adiabatic. The governing equations are solved using the finite-volume method with the SIMPLE algorithm. The numerical procedure adopted in this study yields a consistent performance over a wide range of parameters that were 10⁻⁴ ≤ Da ≤ 10⁻¹ and 0 ≤ Ra _I ≤ 10⁴. The effects of the parameters involved on the heat transfer characteristics are studied in detail. It is found that the variation of the average Nusselt number is non-linear for increasing values of the Darcy number with uniform or non-uniform heating condition.     

Numerical study of transient laminar natural convection heat transfer over a sphere subjected to a constant heat flux

Transient laminar natural convection over a sphere which is subjected to a constant heat flux has been studied numerically for high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and a wide range of Prandtl numbers (Pr = 0.02, 0.7, 7, and 100). A plume with a mushroom-shaped cap forms above the sphere and drifts upward continuously with time. The size and the level of temperature of the transient cap and plume stem decrease with increasing Gr and Pr. Flow separation and an associated vortex may appear in the wake of the sphere depending on the magnitude of Gr and Pr. A recirculation vortex which appears and grows until “steady state” is attained was found only for the very high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and the lowest Prandtl number considered (Pr = 0.02). The appearance and subsequent disappearance of a vortex was observed for Gr = 10⁹ and Pr = 0.7. Over the lower hemisphere, the thickness of both the hydrodynamic (δ_H) and the thermal (δ_T) boundary layers remain nearly constant and the sphere surface is nearly isothermal. The surface temperature presents a local maximum in the wake of the sphere whenever a vortex is established in the wake of the sphere. The surface pressure recovery in the wake of the sphere increases with decreasing Pr and with increasing Gr. For very small Pr, unlike forced convection, the ratio δ_T/δ_H remains close to unity. The results are in good agreement with experimental data and in excellent agreement with numerical results available in the literature. A correlation has also been presented for the overall Nusselt number as a function of Gr and Pr.     

Computational modeling of biomagnetic micropolar blood flow and heat transfer in a two-dimensional non-Darcian porous medium

We study theoretically and computationally the incompressible, non-conducting, micropolar, biomagnetic (blood) flow and heat transfer through a two-dimensional square porous medium in an (x,y) coordinate system, bound by impermeable walls. The magnetic field acting on the fluid is generated by an electrical current flowing normal to the x–y plane, at a distance l beneath the base side of the square. The flow regime is affected by the magnetization B ₀ and a linear relation is used to define the relationship between magnetization and magnetic field intensity. The steady governing equations for x-direction translational (linear) momentum, y-direction translational (linear) momentum, angular momentum (micro-rotation) and energy (heat) conservation are presented. The energy equation incorporates a special term designating the thermal power per unit volume due to the magnetocaloric effect. The governing equations are non-dimensionalized into a dimensionless (ξ,η) coordinate system using a set of similarity transformations. The resulting two point boundary value problem is shown to be represented by five dependent non-dimensional variables, f _ξ  (velocity), f _η (velocity), g (micro-rotation), E (magnetic field intensity) and θ (temperature) with appropriate boundary conditions at the walls. The thermophysical parameters controlling the flow are the micropolar parameter (R), biomagnetic parameter (N _H ), Darcy number (Da), Forchheimer (Fs), magnetic field strength parameter (Mn), Eckert number (Ec) and Prandtl number (Pr). Numerical solutions are obtained using the finite element method and also the finite difference method for Ec=2.476×10⁻⁶ and Prandtl number Pr=20, which represent realistic biomagnetic hemodynamic and heat transfer scenarios. Temperatures are shown to be considerably increased with Mn values but depressed by a rise in biomagnetic parameter (N _H ) and also a rise in micropolarity (R). Translational velocity components are found to decrease substantially with micropolarity (R), a trend consistent with Newtonian blood flows. Micro-rotation values are shown to increase considerably with a rise in R values but are reduced with a rise in biomagnetic parameter (N _H ). Both translational velocities are boosted with a rise in Darcy number as is micro-rotation. Forchheimer number is also shown to decrease translational velocities but increase micro-rotation. Excellent agreement is demonstrated between both numerical solutions. The mathematical model finds applications in blood flow control devices, hemodynamics in porous biomaterials and also biomagnetic flows in highly perfused skeletal tissue. Dedicated to Professor Y.C. Fung (1919-), Emeritus Professor of Biomechanics, Bioengineering Department, University of California at San Diego, USA for his seminal contributions to biomechanics and physiological fluid mechanics over four decades and his excellent encouragement to the authors, in particular OAB, with computational biofluid dynamics research.     

The Onset of Darcy–Brinkman Electroconvection in a Dielectric Fluid Saturated Porous Layer

The combined effect of a vertical AC electric field and the boundaries on the onset of Darcy–Brinkman convection in a dielectric fluid saturated porous layer heated either from below or above is investigated using linear stability theory. The isothermal bounding surfaces of the porous layer are considered to be either rigid or free. It is established that the principle of exchange of stability is valid irrespective of the nature of velocity boundary conditions. The eigenvalue problem is solved exactly for free–free (F/F) boundaries and numerically using the Galerkin technique for rigid–rigid (R/R) and lower-rigid and upper-free (F/R) boundaries. It is observed that all the boundaries exhibit qualitatively similar results. The presence of electric field is emphasized on the stability of the system and it is shown that increasing the AC electric Rayleigh number R _ea is to facilitate the transfer of heat more effectively and to hasten the onset of Darcy–Brinkman convection. Whereas, increase in the ratio of viscosities Λ and the inverse Darcy number Da ⁻¹ is to delay the onset of Darcy–Brinkman electroconvection. Besides, increasing R _ea and Da ⁻¹ as well as decreasing Λ are to reduce the size of convection cells.     

Average and extremal properties of heat transfer and shear stress on a wall surface in Rayleigh–Bénard convection

In this study surface-averaged and extremal properties of heat transfer and shear stress on the upper wall surface of Rayleigh–Bénard convection are numerically examined. The Prandtl number was raised up to 10³, and the Rayleigh number was changed between 10⁴ and 10⁷. As a result, average Nusselt number Nu and shear rate τ/Pr depends on Pr, Ra, and the entire numerical results are distributed between two correlation equations corresponding to small and large Pr. The small and large Pr equations are closely related to steady and unsteady flow regimes, respectively. Nevertheless, a single relation τ/Pr ~ Nu ^3.0 exists to explain the entire results. Similarly the change of local maximal properties Nu _max and τ _max/Pr depends on Pr, Ra, and these values are also distributed between two correlation equations corresponding to small and large Pr cases. Despite such complicated dependence we can obtain a correlation equation as a form of τ _max/Pr ~ Nu _max^2.6, which has not been obtained theoretically.     

The optimal dimensions of convective-radiating circular fins

The optimal dimensions of convective-radiating circular fins with variable profile, heat-transfer coefficient and thermal conductivity, as well as internal heat generation are obtained. A profile of the form y=(w/2) [1+(r _o/r) ⁿ ] is studied, while variation of thermal conductivity is of the form k=k _o[1+ɛ((T−T _∞)/ (T _b−T _∞)) ^m ]. The heat-transfer coefficient is assumed to vary according to a power law with distance from the bore, expressed as h=K[(r−r _o)/(r _e−r _o)]^λ. The results for λ=0 to λ=1.9, and −0.4≤ɛ≤0.4, have been expressed by suitable dimensionless parameters. A correlation for the optimal dimensions of a constant and variable profile fins is presented in terms of reduced heat-transfer rate. It is found that a (quadratic) hyperbolic circular fin with n=2 gives an optimum performance. The effect of radiation on the fin performance is found to be considerable for fins operating at higher base temperatures, whereas the effect of variable thermal conductivity on the optimal dimensions is negligible for the variable profile fin. It is also observed, in general, that the optimal fin length and the optimal fin base thickness are greater when compared to constant fin thickness. Received on 22 February 1999     

Study of airborne particles produced by normal impact of millimetric droplets onto a liquid film

The aim of this work is to carry out an experimental investigation into the generation of airborne microparticles when millimetric droplets of aqueous solutions impact onto a liquid film. Impact experiments using 3.9 mm diameter droplets were carried out for Weber numbers between 159 and 808, with a fixed Ohnesorge number of 2 × 10⁻³ and film parameters S _f (the ratio between the thickness of the liquid film h _film and the diameter of the impacting droplet d _i) between 0.3 and 1. Observed results show that the deposition/splashing threshold is independent of the parameter S _f in agreement with the data in the literature. The aerosol measurement results demonstrate the production of solid particles from the evaporation of secondary microdroplets with diameters less than 30 μm formed when splash occurs. The median diameter of these microdroplets is around 20 μm, corresponding to a value of d ₅₀/d _i = 5 × 10⁻³. Taken together, the results show that the mass and the number of particles emitted increase as the Weber number increases. Moreover, at a Weber number of 808, the results show that the mass and number of particles emitted increases as the parameter S _f decreases. In this case, the mean number of microdroplets emitted per impact is equal to 14 for S _f = 1 and equal to 76 for S _f = 0.3.     

The laminar hole pressure for Newtonian fluids

For flows with wall turbulence the hole pressure, P _H , was shown empirically by Franklin and Wallace (J Fluid Mech, 42, 33–48, 1970) to depend solely on R ₊, the Reynolds number constructed from the friction velocity and the hole diameter b. Here this dependence is extended to the laminar regime by numerical simulation of a Newtonian fluid flowing in a plane channel (gap H) with a deep tap hole on one wall. Calculated hole pressures are in good agreement with experimental values, and for two hole sizes are well represented by: (P _H −P _HS )/τ _w = √(k ² + c ² R ₊²)−k, where the Stokes hole pressure P _HS /τ _w = s (b/H)³, k, c, s are fitted constants, and τ _w is the wall shear stress.     

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.     

Effect of the Source Term on Steady Free Convection Boundary Layer Flows over an Vertical Plate in a Porous Medium. Part I

The Darcy free convection boundary layer flow over a vertical flat plate is considered in the presence of volumetric heat generation/absorption. In the present first part of the paper it is assumed that the heat generation/absorption takes place in a self-consistent way, the source term q ^′′′≡ S of the energy equation being an analytical function of the local temperature difference T − T _∞. In a forthcoming second part, the case of the externally controlled source terms S = S(x,y ) will be considered. It is shown that due to the presence of S, the physical equivalence of the up- and downflows gets in general broken, in the sense that the free convection flow over the upward projecting hot plate (“upflow”) and over its downward projecting cold counterpart (“downflow”) in general become physically distinct. The consequences of this circumstance are examined for different forms of S. Several analytical solutions are given. Some of them describe algebraically decaying boundary layers which can also be recovered as limiting cases of exponentially decayingones. This asymptotic phenomenon is discussed in some detail.     

Conjugate heat transfer study of incompressible turbulent offset jet flows

In the present case, the conjugate heat transfer involving a turbulent plane offset jet is considered. The bottom wall of the solid block is maintained at an isothermal temperature higher than the jet inlet temperature. The parameters considered are the offset ratio (OR), the conductivity ratio (K), the solid slab thickness (S) and the Prandtl number (Pr). The Reynolds number considered is 15,000 because the flow becomes fully turbulent and then it becomes independent of the Reynolds number. The ranges of parameters considered are: OR = 3, 7 and 11, K = 1–1,000, S = 1–10 and Pr = 0.01–100. High Reynolds number two-equation model (k–ε) has been used for turbulence modeling. Results for the solid–fluid interface temperature, local Nusselt number, local heat flux, average Nusselt number and average heat transfer have been presented and discussed.     

Flexural fatigue behavior of threaded connections for large diameter pipes

Full-scale flexural fatigue tests were conducted to investigate the fatigue behavior of a patented threaded connection for large diameter (0.61 m (24 in) outside diameter, 25.4 mm (1 in) wall thickness) offshore pipes. Fifteen fatigue tests were performed by subjecting the threaded connection to constant amplitude stress ranges (between 69 MPa (10 ksi) and 151.8 MPa (22 ksi) on gross cross section) with zero mean stress. The corresponding measured fatigue lives varied from 45000 to 4852200 cycles. Fatigue failures were in the form of cracks through the thickness of the wall and located at the root of the first full contact thread. The failure surfaces were ‘typical’ with identifiable zones of crack initiation, propagation and fracture. Linear regression analysis of the experimental results, namely the applied stress range (S _r ) and the measured number of cycles to failure (N) data, in the log-log domain gave anR ² value of 0.88 and the least-squares best fit equation asS _r (MPa)=1573.2N ^−0.212. The 90% probable fatigue strength prediction equation was estimated asS _r =1393.8N ^−0.212. This equation is recommended for design purposes.     

Experimental and numerical study on laminar natural convection in a cavity heated from bottom due to an inclined fin

Natural convection heat transfer in an inclined fin attached square enclosure is studied both experimentally and numerically. Bottom wall of enclosure has higher temperature than that of top wall while vertical walls are adiabatic. Inclined fin has also adiabatic boundary conditions. Numerical solutions have been done by writing a computer code in Fortran platform and results are compared with Fluent commercial code and experimental method. Governing parameters are Rayleigh numbers (8.10⁵ ≤ Ra ≤ 4 × 10⁶) and inclination angle (30° ≤ and ≤ 120°). The temperature measurements are done by using thermocouples distributed uniformly at the wall of the enclosure. Remarkably good agreement is obtained between the predicted results and experimental data. A correlation is also developed including all effective parameters on heat transfer and fluid flow. It was observed that heat transfer can be controlled by attaching an inclined fin onto wall.     

Precursor ionization ahead of strong shock waves in argon

The mechanism of precursor ionization ahead of strong shock waves has been studied in a low density shock tube. The experimental results are illustrated with Arrhenius plots with kink points dividing them into two parts with apparent activation energy ratio 1:2, namely with the values 7.7 eV and 15.3 eV, and varying with first and third power of the density respectively. A model is proposed to interpret the facts where the process taking place in the precursor region, is a two step photo ionization accompanied with the drift flow effect of the gas relative to the shock wave or the ionization recombination effect according to whether the shock speed and initial density are low enough. The product of the A-A collision excitation cross section coefficientS ^* multiplied by the radiation cross sectionQ ^* of ArgonS ^*×Q ^*=1×10⁻³⁶ (cm⁴eV⁻¹) and the three body recombination coefficient of Argon at room temperaturek _ra =1×10⁻²⁴ (cm⁻⁶s⁻¹). The project supported by the National Natural Science Foundation of China     

ON THE CRITICAL POINTS OF THE MAP Fp:X→‖AXB-C‖pp

Suppose A,B and C are the bounded linear operators on a Hilbert space H, when A has a generalized inverse A^- such that (AA^-)^*=AA^- and B has a generalized inverse B^- such that (B^-B)^*=B^-B,the general characteristic forms for the critical points of the map F_p:X^→‖AXB-C‖_p^p(1p=2. Similarly, the same question has been discussed for several operators.     

Floquet’s Theorem and Stability of Periodic Solitary Waves

This paper is concerned with the spectrum the Hill operator L(y) = −y′′ + Q(x) y in L²_per[0, p]{L^{2}_{\rm per}[0, \pi]} . We show that the eigenvalues of L can be characterized by knowing one of its eigenfunctions. Applications are given to nonlinear stability of a class of periodic problems.     

Experimental study on the heat transfer and pressure drop of a cross-flow heat exchanger with different pin–fin arrays

In this study, convective heat transfer and pressure drop in a cross-flow heat exchanger with hexagonal, square and circular (HSC) pin–fin arrays were studied experimentally. The pin–fins were arranged in an in-line manner. For the applied conditions, the optimal spacing of the pin–fin in the span-wise and stream-wise directions has been determined. The variable parameters are the relative longitudinal pitch (S _L /D = 2, 2.8, 3.5), and the relative transverse pitch was kept constant at S _T /D = 2. The performances of all pin–fins were compared with each other. The experimental results showed that the use of hexagonal pin–fins, compared to the square and circular pin–fins, can lead to an advantage in terms of heat transfer enhancement. The optimal inter-fin pitches are provided based on the largest Nusselt number under the same pumping power, while the optimal inter-fin pitches of hexagonal pin–fins are S _T /D = 2 and S _L /D = 2.8. Empirical equations are derived to correlate the mean Nusselt number and friction coefficient as a function of the Reynolds number, pin–fin frontal surface area, total surface area, and total number. Consequently, the general empirical formula is given in the present form.