The high accuracy explicit difference scheme for solving parabolic equations 3-dimension

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Stability of an Initially, Stably Stratified Fluid Subjected to a Step Change in Temperature

The onset of convective instability in an initially quiescent, stably stratified fluid layer between two horizontal plates is analyzed with linear theory. The bottom boundary is heated suddenly from below, subjected to a step change in surface temperature. The critical time t _c to mark the onset of Rayleigh-Bénard convection is predicted by propagation theory. This theory uses the length scaled by , where α denotes thermal diffusivity. Under the normal mode analysis the dimensionless disturbance equations are obtained as a function of τ(=αt/d ²) and ζ(=Z/), where d is the fluid layer depth and Z is the vertical distance. The resulting equations are transformed to self-similar ones by using scaling and finally fixing τ as τ_c under the frame of coordinates τ and ζ. For a given γ, Pr and τ_c, the minimum value of Ra is obtained from the marginal stability curve. Here γ denotes the temperature ratio to represent the degree of stabilizing effect, Pr is the Prandtl number and Ra is the Rayleigh number. With γ=0, the minimum Ra value approaches the well-known value of 1708 as τ_c increases. However, it is inversely proportional to τ_c ^3/2 as τ_c decreases. With increasing γ, the system becomes more stable. It is interesting that in the present system, propagation theory produces the stability criteria to bound the available experimental data over the whole domain of time. Received 5 November 2001 and accepted 29 March 2002 Published online: 2 October 2002 RID="*" ID="*" This work has been supported by both SK Chemicals Co. Ltd. and LG Chemical Ltd., Seoul under the Brain Korea 21 Project of the Ministry of Education. Communicated by H.J.S. Fernando     

Curing effects on the acoustical properties of epoxy polymers

This paper presents an experimental method for measuring the attenuation and the velocity of longitudinal ultrasonic waves propagating through flat epoxy polymer samples. The study takes place in the first phase of epoxy polymer's polymerization, where these materials pass slowly from liquid state to the solid state. For this purpose an experimental setup was introduced, suitable for the accurate evaluation of the acoustic properties Δα andc _ℓ ^e , when the epoxy polymers are in their first phase of polymerization, while they are cured for 24 hours at room temperature (20°C). The ultrasonic method used is the pulse echo-through transmission technique. From the variation ofc _ℓ ^e and Δα during the first phase of epoxy polymers curing, the three characteristic states: liquid, semi-solid and solid, are clearly determined. It is also observed that plasticizer reduces the testability and the semi-solid state shows greater attenuation than either the liquid or the solid state.     

Scaling law in saddle-node bifurcations for one-dimensional maps: a complex variable approach

The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations, where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium, found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, τ, scales according to an inverse square-root power law, τ∼(μ−μ _c )^−1/2, as the bifurcation parameter μ, is driven further away from its critical value, μ _c . In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling laws of one-dimensional discrete dynamical systems with saddle-node bifurcations.     

The application of Moire interferometry in the measurement of displacement field and strain field at notch-tip and crack-tip

This paper presents the application of Moire interferometry in measuring the displacement and strain field at notch-tip and crack-tip before and after crack propagation. The experiment is carried out using a three point bending beam with a notch. TheN _x andN _y fringe patterns representing displacement field, and the ΔN _x/Δx and ΔN _γ/Δγ fringe patterns representing the strain field are obtained. The sensitivity of the measured displacement is 0.417 μm per fringe order. The displacement and strain distribution along the sectionx=0 have been worked out according toN _x andN _γ fringe patterns. The project supported by Chinese Academy of Sciences and National Natural Science Foundation of China     

Turbulence Modeling Effects on the Prediction of Equilibrium States of Buoyant Shear Flows

The effects of turbulence modeling on the prediction of equilibrium states of turbulent buoyant shear flows were investigated. The velocity field models used include a two-equation closure, a Reynolds-stress closure assuming two different pressure-strain models and three different dissipation rate tensor models. As for the thermal field closure models, two different pressure-scrambling models and nine different temperature variance dissipation rate ɛ_τ) equations were considered. The emphasis of this paper is focused on the effects of the ɛ_τ-equation, of the dissipation rate models, of the pressure-strain models and of the pressure-scrambling models on the prediction of the approach to equilibrium turbulence. Equilibrium turbulence is defined by the time rate of change of the scaled Reynolds stress anisotropic tensor and heat flux vector becoming zero. These conditions lead to the equilibrium state parameters, given by /ɛ, _τ/ɛ_τ, , Sk/ɛ and G/ɛ, becoming constant. Here, and _τ are the production of turbulent kinetic energy k and temperature variance , respectively, ɛ and ɛ_τ are their respective dissipation rates, R is the mixed time scale ratio, G is the buoyant production of k and S is the mean shear gradient. Calculations show that the ɛ_τ-equation has a significant effect on the prediction of the approach to equilibrium turbulence. For a particular ɛ_τ-equation, all velocity closure models considered give an equilibrium state if anisotropic dissipation is accounted for in one form or another in the dissipation rate tensor or in the ɛ-equation. It is further found that the models considered for the pressure-strain tensor and the pressure-scrambling vector have little or no effect on the prediction of the approach to equilibrium turbulence. Received 21 April 2000 and accepted 21 February 2001     

Axial annular flow of plastic fluids: dead zones and plug-free flow

In axial annular flow, the shear stress decreases from its value τ(κR) at the inner cylinder to 0 at r = λR and increases from then on to τ(R) at the outer cylinder. For plastic fluids with a yield stress τ _c, λ will be such that flow commences when τ(κR) = τ(R) = τ _c. For fluids with position-dependent yield stresses (electro- and magnetorheological fluids are examples), the situation is more complex. While it is possible that yielding and flow occur everywhere, it is also possible that flow occurs only in parts of the fluid-filled space, and a dead zone (region in which the fluid is at rest) close to one of the walls exists. In that case, the fluid will flow no matter how small the applied pressure difference is. If P is large enough, the dead zone ceases to exist and flow without any plug is possible. The fluid flows as if no yield stress exists.

Near-wall hot-wire measurements

This work continues the studies of Khoo et al. (Exp. Fluids 29: 448–460, 2001), where experiments were performed in turbulent-channel and flat-plate boundary-layer flows using near-wall hot-wire probes. The probability density function (pdf) of the wall-shear stress and streamwise velocity fluctuations in the viscous sublayer, buffer region and beyond were compared and analyzed. The convective velocity U _c of the streamwise velocity fluctuations in the very near-wall region was obtained using a two-point correlation technique. It was found that in the viscous sublayer, U _c is approximately constant at 13u _τ and 15u _τ , respectively, for the channel and boundary-layer flows. Spectra data for the viscous sublayer are presented for the first time, and the normalized spectral plots for different flow conditions collapse at high frequencies or wavenumbers, thus indicating the possible presence of small-scale universality at different Reynolds numbers. The integral time scale corresponding to the streamwise velocity fluctuations in the viscous sublayer is also presented. Received: 18 October 2000/Accepted: 2 April 2001     

Further Study on Pure Shear

It is known that the Cauchy stress tensor T is a pure shear when trT = 0. An elementary derivation is given for a coordinate system such that, when referred to this coordinate system, the diagonal elements of T vanish while the off-diagonal elements τ ₁, τ ₂, τ ₃, are the pure shears. The structure of τ _i (i = 1, 2, 3) depends on one non-dimensional parameter q = 54(detT)² / [tr(T ²)]³, 0 ≤ q ≤ 1. When q = 0, one of the three τ _i vanishes. A coordinate system can be chosen such that the remaining two have the same magnitude or one of the remaining two also vanishes. When q = 1, all three τ _i have the same magnitude. However, there is a one-parameter family of coordinate systems that gives the same three τ _i . For q ≠ 0 or 1, none of the three τ _i vanishes and the three τ _i in general have different magnitudes. Nevertheless, a coordinate system can be chosen such that two of the three τ _i have the same magnitude. ^★Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.     

Effect of a concentrated mass on coriolis flowmetering

Summary Usually a Coriolis mass flowmeter consists of a fluid conveying vibrating pipe segment with a reflection symmetry, on which the time delay is measured between the vibrations of two symmetrically situated cross sections. For a homogeneous pipe segment, the proportionality factorK _c between and the mass flowrate , i.e. the calibration factor of the instrument, is independent of the nature of the flowing fluid. Fixing a concentrated massm _c at the middle of the pipe segment — as required e.g. for the purpose of a symmetric excitation of the vibration — brings about a dependence of the factorK _c on the fluid density. In the present paper the influence of the massm _c on the vibration spectrum and on flowmetering is investigated in detail for an instrument working with a straight pipe segment. It turns out that, whereas the frequency of the fundamental vibration mode is strongly influenced bym _c , the calibration factorK _c is practically independent of the massm _c , up to fairly high values compared to the mass of the fluid filled pipe segment.

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Effekt einer konzentrierten Masse auf die Coriolis-Durchflußmessung

Übersicht Ein Coriolis-Durchflußmeßgerät ist üblicherweise ein von einem Fluid durchströmter schwingender Rohrabschnitt, der eine Spiegelungssymmetrie besitzt und an dem der Zeitunterschied zwischen den Schwingungen zweier symmetrisch gelegener Querschnitte gemessen wird. Für einen homogenen Rohrabschnitt ist der ProportionalitätsfaktorK _C zwischen und dem Massenstrom , d. h. der Kalibrierungsfaktor des Instrumentes, unabhängig von der Natur des Fluids. Das Anbringen einer konzentrierten Massem _c an der Mitte des Rohrabschnittes — etwa zum Zwecke einer symmetrischen Anregung der Schwingung — hat eine Abhängigkeit des FaktorsK _C von der Fluiddichte zur Folge. In der vorliegenden Arbeit wird der Einfluß der Massem _c auf das Schwingungsspektrum und die Durchflußmessung bei einem Instrument mit geradem Rohrabschnitt eingehend untersucht. Es ergibt sich, daß die Schwingungsfrequenz der Grundmode zwar stark vonm _c beeinflußt wird, der KalibrierungsfaktorK _C aber praktisch unabhängig vonm _c ist, bis zu ziemlich hohen Werten vonm _c verglichen mit der Masse des gefüllten Rohrabschnittes.

     

Onset of Buoyancy-Driven Convection in Porous Media Saturated with Cold Water Cooled from Above

When porous media saturated with initially stagnant cold water around the density maximum temperature are cooled from above, convection may be induced in an unstable lower layer. In this study, the onset of buoyancy-driven convection during time-dependent cooling is investigated using the propagation theory, which transforms disturbance equations similarly, and also considering the density inversion effect. The critical Darcy–Rayleigh number Ra _D,c is found as a function of the dimensionless density maximum temperature θ _max. For Ra _D > Ra _D,c the dimensionless critical time τ _c to mark the onset of instability is presented as a function of Ra _D and θ _max. These critical conditions are compared with previous theoretical results.     

A photoelastic study of crack repair

Birefringent coatings have been employed to study the effectiveness of an adhesively bonded repair of a center-cracked tension panel. The repair was one sided, with photoelastic coatings applied to the opposite side. Photoelastic coatings were also applied over the patch. Analysis methods are presented to permit the stress intensity factor to be determined from the isochromatic fringe patterns recorded from both continuous andX- andY-edged coatings. The results showed that the one-sided adhesively bonded patch reduced the stress intensity factor; however, the repair did not markedly change the character of stress distributions. Fringe loops formed near the crack tips for both the cracked and repaired tension panels. The primary difference was in the size of the loops. The reduction inK _I due to repair was smaller than anticipated, but even small improvements in ΔK _I markedly enhance the life of a repaired panel. The Paris power law is used to show the relation between the reduction in ΔK _I and the improvement in the crack growth rateda/dN. Fringe patterns from the birefringent coatings applied to the patch provided a means not only to investigate the stresses in the patch but also to detect the initiation of the local debonding of the adhesive in the neighborhood of the crack. The birefringent coating on the patch is an approach for producing an optically “smart” repair.     

On the multiple scales perturbation method for difference equations

In the classical multiple scales perturbation method for ordinary difference equations (O Δ Es) as developed in 1977 by Hoppensteadt and Miranker, difference equations (describing the slow dynamics of the problem) are replaced at a certain moment in the perturbation procedure by ordinary differential equations (ODEs). Taking into account the possibly different behavior of the solutions of an O Δ E and of the solutions of a nearby ODE, one cannot always be sure that the constructed approximations by the Hoppensteadt–Miranker method indeed reflect the behavior of the exact solutions of the O Δ Es. For that reason, a version of the multiple scales perturbation method for O Δ Es will be presented and formulated in this paper completely in terms of difference equations. The goal of this paper is not only to present this method, but also to show how this method can be applied to regularly perturbed O Δ Es and to singularly perturbed, linear O Δ Es.     

A Similarity Subgrid Model for Premixed Turbulent Combustion

The accuracy of large-eddy simulation (LES) of a turbulent premixed Bunsen flame is investigated in this paper. To distinguish between discretization and modeling errors, multiple LES, using different grid sizes h but the same filterwidth Δ, are compared with the direct numerical simulation (DNS). In addition, LES using various values of Δ but the same ratio Δ/h are compared. The chemistry in the LES and DNS is parametrized with the standard steady premixed flamelet for stochiometric methane-air combustion. The subgrid terms are closed with an eddy-viscosity or eddy-diffusivity approach, with an exception of the dominant subgrid term, which is the subgrid part of the chemical source term. The latter subgrid contribution is modeled by a similarity model based upon 2Δ, which is found to be superior to such a model based upon Δ. Using the 2Δ similarity model for the subgrid chemistry the LES produces good results, certainly in view of the fact that the LES is completely wrong if the subgrid chemistry model is omitted. The grid refinements of the LES show that the results for Δ = h do depend on the numerical scheme, much more than for h = Δ/2 and h = Δ/4. Nevertheless, modeling errors and discretization error may partially cancel each other; occasionally the Δ = h results were more accurate than the h ≤ Δ results. Finally, for this flame LES results obtained with the present similarity model are shown to be slightly better than those obtained with standard β-pdf closure for the subgrid chemistry.     

Unconditional nonlinear exponential stability of the motionless conduction-diffusion solution

Nonlinear stability of the motionless state of a heterogeneous fluid with constant temperature-gradient and concentration-gradient is studied for both cases of stress-free and rigid boundary conditions. By introducing new energy functionals we have shown that for τ=P _C /P _T ≤1, the motionless state is always stable and for τ≤1, the sufficient and necessary conditions for stability coincide, whereP _C ,P _T ,C andR are the Schmidt number, Prandtl number, Rayleigh number for solute and heat respectively. Moreover, the criteria guarantees the exponential stability. The Project supported by SRF for ROCS, China Postdoctoral Science Foundation and the National Basic Research Project “Nonlinear Science”     

Asymptotics of the Fast Diffusion Equation via Entropy Estimates

We consider non-negative solutions of the fast diffusion equation u _t  = Δ u ^m with m ∈ (0, 1) in the Euclidean space , d ≧ 3, and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t → ∞ for m ≧ m _c  = (d − 2)/d, or as t approaches the extinction time when m < m _c . For a class of initial data, we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m ≧ m _c , or close enough to the extinction time if m < m _c . Such results are new in the range m ≦ m _c where previous approaches fail. In the range m _c  < m < 1, we improve on known results.     

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.     

Influence of mixture sensitivity and pore size on detonation velocities in porous media

Experiments have been carried out to determine the dependence of the detonation velocity in porous media, on mixture sensitivity and pore size. A detonation is established at the top end of a vertical tube and allowed to propagate to the bottom section housing the porous bed, comprised of alumina spheres of equal diameter (1–32 mm). Several of the common detonable fuels were tested at atmospheric initial pressure. Results indicate the existence of a continuous range of velocities with change in Φ, spanning the lean and the rich propagation limits. For all fuels in a given porous bed, the velocity decreases from a maximum value at the most sensitive mixture near Φ≈1 (minimum induction length), toV/V _CJ≈0.3 at the limits. A decrease in pore size brings about a reduction inV/V _CJ and a narrowing of the detonability range for each fuel. For porous media comprised of spherical particles, it was possible to correlate the velocity data corresponding to a variety of different mixtures and for a broad range of particle sizes, using the following empirical expression:V/V _CJ=[1–0.35 log(d _c /d _p)]±0.1. The critical tube diameterd _c is used as a measure of mixture sensitivity andd _p denotes the pore diameter. An examination of the phenomenon at the composition limits, suggests that wave failure is controlled by a turbulent quenching mechanism.     

Non-linear rheology and flow-induced transition of a lamellar-to-vesicle phase in ternary systems of alkyldimethyl oxide/alcohol/water

 The time-dependent transformation of an ionically charged lamellar phase (L _α-phase) into a vesicle phase under the influence of shear is investigated using rheological and conductivity measurements. The L _α-phase consists of the zwitterionic surfactant tetradecyldimethylaminoxide (C₁₄DMAO), hexanol, oxalic acid and water. The experiments were carried out on the L _α-phase in a well defined state. It was prepared by a special route from the neighbouring L ₃-phase that consists of 100 mM C₁₄DMAO, 250 mM hexanol and 5 mM oxalicdiethylester (OEE). The OEE hydrolyses in the L ₃ -phase to oxalic acid and ethanol. The result is a virgin L _α-phase which consists of stacked bilayers and which has not been exposed to shear. When this low-viscous phase is subjected to shear it is transformed into a highly viscous vesicle phase. The transformation of the L _α-phase into vesicles under constant shear was monitored by recording the viscosity and conductivity with time. It is observed that at least three different time constants can be distinguished in the transformation process. The conductivity passes through a minimum (τ₁) in the direction of shear. The viscosity first passes through a minimum (τ₂) and then over a maximum (τ₃). It is concluded that τ₁ belongs to the complete alignment of the bilayer parallel to the wall, τ₂ to the beginning of the break-up of the bilayers to the vesicles and τ₃ to the complete transformation of the L _α- to the vesicle phase. When the shear rate was varied, it was noted that the product of the time constants and shear is constant. Received: 30 June 1999/Accepted: 30 August 1999