Some basic problems in numerically simulating effective properties and local fields of composite materials

It is pointed out that to numerically estimate the effective properties and local fields of matrix-inclusion composites, a commonly adopted method is accompanied with some serious draw-backs. We call this method the nominal loading scheme (NLS), which considers the actual inclusion distribution inside a finite domain, Ω say, treats the external domain of Ω to be of the pure matrix material, and imposes the actural traction, σ^∞ say on the remote boundary. It thus gives rise to the following basic problems: (i) Can NLS be improved remarkably just by adjusting σ^∞? (ii) What is the relationship between the size of Ω and the scale of inclusions? (iii) Which choice is better in calculating the effective properties, the whole domain Ω or an appropriately selected sub-domain of Ω? Targeting these problems, the equivalent loading scheme (ELS) and equivalent matrix scheme (EMS) are proposed. It is theoretically analyzed that both ELS and EMS can be used to precisely simulating the effective properties and local fields of matrix-inclusion composities, and both ELS and EMS are self-approved. As an application, ELS combined with a so-called pseudo-dislocations method is used to evaluate the effective properties and local fields of two-dimensional two-phase composites with close-packed circular inclusions, or randomly distributed circular inclusions, or randomly distributed microcracks. The results show that substituting the remore traction σ^∞ with the effective stress field σ^ɛ suggested by IDD scheme is a simple and effective method, and the estimation of the effective properties and local fields is very close to the accurate solution. The present work is supported by the National Natural Science Foundation of China (No. 19525207).     

Effect of local forcing on a turbulent boundary layer

An experimental study is performed to analyze flow structures behind local suction and blowing in a flat-plate turbulent boundary layer. The local forcing is given to the boundary layer flow by means of a sinusoidally oscillating jet issuing from a thin spanwise slot at the wall. The Reynolds number based on the momentum thickness is about Re _θ =1700. The effects of local forcing are scrutinized by altering the forcing frequency (0.011 ≤ f ⁺≤ 0.044). The forcing amplitude is fixed at A ₀=0.4. It is found that a small local forcing reduces the skin friction and the skin friction reduction increases with the forcing frequency. A phase-averaging technique is employed to capture the large-scale vortex evolution. An organized spanwise vortical structure is generated by the local forcing. The cross-sectional area of vortex and the time fraction of vortex are examined by changing the forcing frequency. An investigation of the random fluctuation components reveals that turbulent energy is concentrated near the center of vortical structures. Received: 17 March 2000/Accepted: 3 April 2001     

Experiments on the lift of a spinning sphere in a range of intermediate Reynolds numbers

 The lift force experienced by a spinning sphere moving in a viscous fluid, with constant linear and angular velocities, is measured by means of a trajectographic technique. Measurements are performed in the range of dimensionless angular velocities γ=aω/V lying between 1 and 6, and in the range of Reynolds numbers Re=2aV/ν lying between 10 and 140 (a sphere radius, ω angular velocity, V relative velocity of the sphere centre, ν fluid kinematic viscosity). A notable departure from the theoretical relationship at low Reynolds number, C _L =2γ, is obtained, the ratio C _L /γ being found to significantly decrease with increasing γ and increasing Re. The following correlation is finally proposed to estimate the lift coefficient in the range 10<Re<140: C _L ≅0.45+(2γ−0.45) exp (−0.075γ^0.4 Re ^0.7) Received: 20 May 1996/Accepted: 9 November 1997     

Effect of porous disc receiver configurations on performance of solar parabolic trough concentrator

In this article, heat transfer enhancement of line focus solar collector with porous disc receiver is studied with water and therminol oil. A three dimensional (3-D) numerical simulation of porous disc enhanced receiver is carried out using commercial CFD software Fluent 6.3 to evolve the optimum configuration. The 3-D numerical model is solved by renormalization-group based k-ε turbulent model associated with standard wall function. The effect of porous disc receiver configurations (solid disc at bottom; porous disc at bottom; porous disc at top; and alternative porous disc) on performance of the trough concentrator is investigated. The effect of porous disc geometric parameters (φ, θ, W, H and t) and fluid parameters (Pr and m) on heat transfer enhancement of the receiver is also studied. The numerical simulation results show that the flow pattern around the solid and porous discs are entirely different and it significantly influences the local heat transfer coefficient. The porous disc receiver experiences low pressure drop as compared to that of solid disc receiver due to less obstruction. The optimum configuration of porous disc receiver enhances the heat transfer rate of 221 W m⁻¹ and 13.5% with pumping penalty of 0.014 W m⁻¹ for water and for therminol oil-55, heat transfer rate enhances of 575 W m⁻¹ and 31.4% with pumping penalty of 0.074 W m⁻¹ as compared to that of tubular receiver at the mass flow rate of 0.5 kg s⁻¹. The Nusselt number and friction factor correlations are proposed for porous disc receiver to calculate heat transfer characteristics. The porous disc receiver can be used to increase the performance of solar parabolic trough concentrator.     

On the Domain of Validity of Brinkman’s Equation

An increasing number of articles are adopting Brinkman’s equation in place of Darcy’s law for describing flow in porous media. That poses the question of the respective domains of validity of both laws, as well as the question of the value of the effective viscosity μ ^e which is present in Brinkman’s equation. These two topics are addressed in this article, mainly by a priori estimates and by recalling existing analyses. Three main classes of porous media can be distinguished: “classical” porous media with a connected solid structure where the pore surface S _p is a function of the characteristic pore size l _p (such as for cylindrical pores), swarms of low concentration fixed particles where the pore surface is a function of the characteristic particle size l _s , and fiber-made porous media at low solid concentration where the pore surface is a function of the fiber diameter. If Brinkman’s 3D flow equation is valid to describe the flow of a Newtonian fluid through a swarm of fixed particles or fibrous media at low concentration under very precise conditions (Lévy 1983), then we show that it cannot apply to the flow of such a fluid through classical porous media.     

Investigation of the different base fluid effects on the nanofluids heat transfer and pressure drop

A numerical study of laminar forced convective flows of three different nanofluids through a horizontal circular tube with a constant heat flux condition has been performed. The effect of Al₂O₃ volume concentration 0 ≤ φ ≤ 0.09 in the pure water, water-ethylene glycol mixture and pure ethylene glycol as base fluids, and Reynolds number of 100 ≤ Re ≤ 2,000 for different power inputs in the range of 10 ≤ Q(W) ≤ 400 have been investigated. In this study, all of the nanofluid properties are temperature and nanoparticle volume concentration dependent. The governing equations have been solved using finite volume approach with the SIMPLER algorithm. The results indicate an increase in the averaged heat transfer coefficient with increasing the mass of ethylene glycol in the water base fluid, solid concentration and Reynolds number. From the investigations it can be inferred that, the pressure drop and pumping power in the nanofluids at low solid volumetric concentration (φ < 3%) is approximately the same as in the pure base fluid in the various Reynolds numbers, but the higher solid nanoparticle volume concentration causes a penalty drop in the pressure. Moreover, this study shows it is possible to achieve a higher heat transfer rate with lower wall shear stress with the use of proper nanofluids.     

Characterization of the flow properties of sodium carboxymethylcellulose via mechanical and optical techniques

Sodium carboxymethylcellulose (NaCMC) in solution represents a complex rheological system, since it forms aggregates and associations and hence higher-level structures and, depending on the synthesis, is only found in a molecularly dispersed form in exceptional cases. Rheo-mechanical investigations of the viscoelasticity showed that the Cox-Merz rule is not fulfilled. The aim was therefore to examine whether rheo-optics could be employed to provide more detailed conclusions about the parameters that influence the flow behavior of NaCMC than has hitherto been available with mechanical methods. The flow birefringence, Δn ^′, rises as the degree of polymerization increases, and exhibits the same dependence on molar mass as does the viscosity: Δn ^′∝M _w ^3.4. As the degree of polymerization increases while the shear rate remains constant, the polymer segments become more distinctly aligned in the direction of shear. Hence increasing the degree of polymerization also affects the solution structure, i.e. the interaction of the molecules with one another. The stress-optical rule only applies to a limited extent for this system. The stress-optical coefficient, C, is almost independent of the shear rate, but is strongly influenced by the concentration and attains a limiting value of 3 × 10⁻⁸ Pa⁻¹. C was determined for a polymer in dilute solution and the curve obtained also enabled transitions in the solution structure to be recognized. Received: 1 May 1998 Accepted: 5 October 1998     

Exact solutions for the flow of second grade fluid in annulus between torsionally oscillating cylinders

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. At time t = 0, the fluid and both the cylinders are at rest and at t = 0 + , cylinders suddenly begin to oscillate around their common axis in a simple harmonic way having angular frequencies ω 1 and ω 2 . The obtained solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for Newtonian fluid are also obtained as limiting cases of our general solutions.     

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.     

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.     

Estimating the viscoelastic moduli of complex fluids using the generalized Stokes–Einstein equation

We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, <Δr ²(t)>, of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G ^*(ω), in the Fourier frequency domain, and the stress relaxation modulus, G _r (t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of <Δr ²(t)> known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing <Δr ²(t)> as a local power law. If the logarithmic slope of <Δr ²(t)> can be accurately determined, these estimates generally perform well at the frequency extremes. Received: 8 September 2000/Accepted: 9 March 2000     

Validation of the thermal equilibrium assumption in the transient conjugated forced convection channel flow

 The validity of the local thermal equilibrium assumption in the transient conjugated forced convection channel flow is investigated analytically. Closed form expressions are presented for the temperatures of the fluid and solid domains and for the criterion which insure the validity of the local thermal equilibrium assumption. It is found that three dimensionless parameters control the local thermal equilibrium assumption. These parameters are the Biot number Bi, the dimensionless channel length ξ_max and the solid to fluid total thermal capacity ratio C _R. The qualitative and quantitative aspects of the effects of these three parameters on the channel thermalization time are investigated. Received on 9 August 2000     

Investigation of the flow around a circular cylinder under the influence of an electromagnetic force

An investigation of the effect of the local electromagnetic body force on the flow behavior around a circular cylinder is conducted. The electromagnetic force is applied locally on the cylinder surface in the range of 70–130° from the stagnation point along the cylinder circumference in both clockwise and counterclockwise directions. The numerical results predict that the Lorentz force applied in the circumferential direction on the cylinder moves the separation point rearward, and reduces the drag. To validate the numerical results, an experiment is conducted with a circular cylinder of 5 cm diameter. The electrodes and permanent magnets are flush mounted on the cylinder in such a way that the Lorentz force is generated in the circumferential direction. Flow visualization with polystyrene particles and direct drag measurement using strain gages are made. The fluid used is natural sea water of electric conductivity of about 4 (Ω m)^-1. Induction effect can be neglected in the present investigation due to the low flow speed and the Lorentz force is proportional to E×B where E is an applied electric field and B is a magnetic field. Received: 7 June 1998/Accepted: 28 April 1999     

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.     

Vortex ring generation due to the coalescence of a water drop at a free surface

 Results are presented of an experimental investigation of vortex ring formation by a fluid drop contacting a free surface with negligible velocity. The pool fluid is mixed with fluorescein dye, and a laser sheet is used to illuminate a plane of the flow. A series of representative images is recorded by a CCD camera and speculation is made regarding specific sources of vorticity flux through the free surface. Two scaling analyses previously presented by other investigators are demonstrated to be equivalent under the assumptions of this experiment, and they provide the motivation for a series of test runs in which the duration of the coalescence process, τ_*, is related to variations in drop diameter L and fluid surface tension σ. Experimental results are in agreement with the analyses, showing τ_*∼σ^-1/2 and τ_*∼L ^3/2. Received: 22 December 1995 / Accepted: 15 October 1996     

Linear viscoelastic behavior of perfluorooctyl sulfonate micelles: effects of counter-ions

Linear viscoelastic behavior was investigated for aqueous solutions of perfluorooctyl sulfonate (C₈F₁₇SO⁻ ₃; abbreviated as FOS) micelles having a mixture of tetraethylammonium (N⁺(C₂H₅)₄; TEA) and lithium (Li⁺) ions as the counter-ions. The solutions had the same FOS concentration (0.1 mol l⁻¹) and various Li⁺ fractions in the counter-ions, φ_Li = 0−0.6, and the FOS micelles in these solutions formed threads which further organized into dendritic networks. At T ≤ 15 °C, the terminal relaxation time τ and the viscosity η, governed by thermal scission of the networks, increased with increasing φ_Li up to 0.55. A further increase of φ_Li resulted in decreases of τ and η and in broadening of the relaxation mode distribution. These rheological changes are discussed in relation to the role of TEA ions in thermal scission: Previous NMR studies revealed that only a fraction of TEA ions were tightly bound to the FOS micellar surfaces and these bound ions stabilized the thread/network structures. The concentration of non-bound TEA ions, C_TEA ^*, decreased and finally vanished on increasing φ_Li up to φ_Li ^* ≅ 0.6, and the concentration of the bound TEA ions significantly decreased on a further increase of φ_Li. The non-bound TEA ions appeared to catalyze the thermal scission of the FOS threads, and the observed increases of τ and η for φ_Li < 0.55 were attributed to the decrease of C_TEA ^*. On the other hand, the decreases of τ and η as well as the broadening of the mode distribution, found for φ_Li > 0.55 (where C_TEA ^* ≅ 0), were related to destabilization of the FOS threads/networks due to a shortage of the bound TEA ions and to the existence of concentrated Li⁺ ions. Viscoelastic data of pure FOSTEA and FOSTEA/FOSLi/TEACl solutions lent support to these arguments for the role of TEA ions in the relaxation of FOSTEA/FOSLi solutions. Received: 12 October 1999/Accepted: 1 November 1999     

Poroelasticity-I: Governing Equations of the Mechanics of Fluid-Saturated Porous Materials

We present here the approach to the theory of fluid-filled poroelastics based on consideration of poroelastics as a continuum of “macropoints” (representative elementary volumes), which “internal” states can be described by as a set of internal parameters, such as local relative velocity of fluid and solid, density of fluid, internal strain tensor, specific area, and position of the center of mass of porous space. We use the generalized Cauchy–Born hypothesis and suggest that there is a system of (structural) relationships between external parameters, describing the deformation of the continuum and internal parameters, characterizing the state of representative elementary volumes. We show that in nonhomogenous (and, particularly, nonlinear) poroelastics, an interaction force between solid and fluid appears. Because this force is proportional to the gradient of porosity, absent in homogeneous poroelastics, and one can neglect with dynamics of internal degrees of freedom, this force is equivalent to the interaction force, introduced earlier by Nikolaevskiy from phenomenological reasons. At last, we show that developed theory naturally incorporates three mechanisms of energy absorption: visco-inertial Darcy mechanism, “squirt flow” attenuation, and skeleton attenuation.     

A NOTE ON BIFURCATIONS OF u″+μ(u-uk)=0(4≤k∈Z+)

Bifurcations of one kind of reaction-diffusion equations, u″+μ(u-u^k)=0(μ is a parameter,4≤k∈Z⁺), with boundary value condition u(0)=u(π)=0 are discussed. By means of singularity theory based on the method of Liapunov-Schmidt reduction, satisfactory results can be acquired.     

Radial filtration during constant-force squeeze flow of soft solids

Various soft solid suspensions were squeezed at constant force between polished and roughened circular glass plates and the time-dependence of the interplate separation was measured. The filterability of suspensions was quantified by their desorptivity S obtained from measurements of capillary suction time. The squeeze flow (SF) of suspensions for which S < 2 μm s^−1/2 was largely consistent with rheological theory, which neglected radial filtration: the relative motion between the liquid and solid phases of the suspension in the radial pressure gradient. Suspensions having S > 2 μm s^−1/2 showed SF behavior that was consistent with the presence of radial filtration.     

Solid transport measurements through image processing

The paper describes a novel technique which enables evaluation of sediment fluxes on the upper layer of a granular bed by means of separate measurements of concentration and velocity of the moving particles. Specific elaboration techniques based on digital image processing were applied to films of the solid fluxes, allowing for automatic measurement. Sediment concentration was measured via a technique based on image subtraction and following proper filtering procedures, while grain velocity was measured by Particle Image Velocimetry. The method was applied in laboratory experiments of one dimensional bed load; the solid discharges measured by the proposed image processing technique were compared to those obtained by manual count of the grains passing over a fixed plate used as a sight. After calibration of the automatic technique, dimensionless solid discharges ranging from 1.0 × 10⁻³ to 1.2 × 10⁻¹ were measured with a maximum error as large as 25%. The technique proposed also enables measurement of the time variation of the quantities and the two dimensional direction of sediment motion, for a complete characterization of grain kinematics in solid transport processes.