Boundary Layer Terms Research Articles

Possible Feedback of Winter Sea Ice in the Greenland and Barents Seas on the Local Atmosphere

Using monthly Arctic sea ice concentration data (1953–95) and the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis dataset (1958–99), possible feedbacks of sea ice variations in the Greenland and Barents Seas to the atmosphere are investigated. Winter (February–April) sea ice anomalies in the Greenland and Barents Seas display important feedback influences on the atmospheric boundary layer in terms of both thermodynamic and dynamic processes. The vertical gradients of potential pseudo-equivalent temperature (θse) between 850 and 700 hPa are greater over sea ice than over open water, implying that a more stable boundary layer forms below 700 hPa over sea ice. The effects of temperature advection are shown to account for a relatively small percentage of the temperature variance in area of sea ice feedbacks. Horizontal and vertical variations of the effects of sea ice on temperature in the Nordic and Barents Seas create the potential for dynamical influences on the atmospheric boundary layer through vertical motion induced by the pressure anomalies in the boundary layer.

Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems

In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergence rate O(N-2 ln2 N + eN-1.5lnN) in a discrete e-weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter e. Numerical tests indicate that the rate O(N-2ln2 N) is sharp for the boundary layer terms. As a by-product, an e-uniform convergence of the same order is obtained for the L2-norm. Furthermore, under the same regularity assumption, an e-uniform convergence of order N-3/2ln5/2N + eN-1ln1/2N in the L∞ norm is proved for some mesh points in the boundary layer region.

Averaging a transport equation with small diffusion and oscillating velocity

AbstractA complete asymptotic expansion is constructed for the transport equation with diffusion term small with respect to the convection. Error estimates are obtained by using matched asymptotic expansion technique and building all the boundary layer terms in time and in space, necessary for obtaining an accurate error estimate. Copyright © 2003 John Wiley & Sons, Ltd.

COUCHE LIMITE DANS UN MODÈLE DE FERROMAGNÉTISME

ABSTRACT In this paper we study the asymptotic limit of the solutions of Landau-Lifschitz equations as the exchange coefficient goes to zero. In order to perform this limit in regular spaces, we have to take into account the singularity induced by a boundary layer term.

Interfacial strength in glass fibre-polypropylene composites: influence of chemical bonding and physical entanglement

For glass fibre–polypropylene (PP) composites, the non-polar nature of polypropylene presents a problem. The present investigation shows that it is necessary to introduce a functionalised PP, for example PP-g-MAH, in order to enhance the bond strength between the PP matrix and aminosilane treated glass fibre. To achieve a better bonding between the substances, three different systems (1–3) in addition to a reference system (0), have been investigated in this study. The two first are based on PP-g-MAH coupling agents, with different concentrations of acid anhydride groups, and the third is a directly reacting system. In the first system, the silane treated glass fibre is exposed to molten mixture of 95 wt% PP homopolymer and 5 wt% PP-g-MAH. In the second system, the silane treated glass fibre is covered by a thin layer of PP-g-MAH and thereafter exposed to the molten PP. The interfacial shear strength is highest for the systems with the pre-compounded graft-copolymer. The resulting influence of the selected coupling systems on the interfacial bond strength of single fibre composite is studied by fragmentation testing. The intermolecular shear strength between fibre and matrix increases with the intermolecular entanglement length of the PP-g-MAH and not by the degree of functionalisation. The PP-g-MAH mixed into the PP gave better results than the route of first covering the glass fibre with a thin layer of PP-g-MAH. This is explained in terms of the probability of generating entanglements and in terms of a weak boundary layer at the glass surface. This conclusion is also supported by the results from using the third principle, i.e. direct reaction between the PP matrix and azidosilane treated glass fibres.

Fundamental correlations for laminar and turbulent free convection from a uniformly heated vertical plate

A dimensionless parameter Π Q is introduced for the correlation of the heat transfer in natural convection induced by a constant wall heat flux. This dimensionless parameter depends on the Prandtl number, Pr, and the modified Rayleigh number, Ra ∗ , as Π Q∼Ra ∗/(1+Pr −1) . The development of Π Q is based on physical arguments and allows the use of a single heat transfer correlation over the entire Prandtl number range. Dimensional arguments in terms of the laminar boundary layer and turbulent microscale concepts lead to a Nu∼ Π Q 1/5 and Nu∼ Π Q 1/4 dependence for the laminar and turbulent heat transfer, respectively. Comparison of these correlations with existing published data shows a good agreement.

Développements asymptotiques dans les coques elliptiques : équations tridimensionnelles linéarisées

We study the asymptotic expansion of the solution of the three-dimensional linearized elasticity equations for a homogeneous and isotropic material in the case of an elliptic shell clamped along its whole lateral face. We show that for smooth data, this solution admits an asymptotic expansion in powers of ε 1/2 where ε represents the (half-)thickness of the shell. This expansion contains boundary layer terms with two scales: terms exponentially decaying with respect to r/ ε where r denotes the geodesic distance to the lateral boundary of the shell, and terms exponentially decaying with respect to r/ε like for plates.

Développements asymptotiques dans les coques elliptiques : Modèle de Koiter

We study the asymptotic expansion of the solution of the two-dimensional Koiter model (see [7]) in linear elasticity, for an elliptic shell clamped along its whole lateral face. We show that for smooth data, this solution admits an asymptotic expansion in powers of ε 1/2 where ε represents the (half-)thickness of the shell. This expansion contains terms independent on ε and boundary layer terms exponentially decaying with respect to r/ ε where r denotes the geodesic distance to the boundary of the mean surface.

Vertical distribution of oxides of nitrogen in the semi-urban planetary boundary layer: mixing ratios, sources and transport

Measurements of the mixing ratios of tropospheric NO and NOY (defined as nitric oxide (NO)+nitrogen dioxide (NO2)+peroxyacetyl nitrate (PAN)+nitric acid (HNO3)+particulate nitrate (NO3−)+⋯) were made over a semi-urban area of central North Carolina at the surface (10 m) and on a tower at heights of 250 m (820 ft) and 433 m (1420 ft) above ground level (AGL) from December 1994 to February 1995. These measurements were compared with synoptic weather data and regional and local upper air soundings in an effort to characterize NO and NOY in the planetary boundary layer in terms of their vertical distributions, diurnal profile, and related transport mechanisms. A pronounced decreasing vertical gradient in both NO and NOY mixing ratios was observed, with a distinct diurnal cycle and nocturnal minimum. Furthermore, the results suggest that NO and NOY were mixed upward from the surface during passage of synoptic meteorological features (and their associated vertical motions). Most importantly, the data reveals that mixing ratios of NO and NOY at the elevated heights did not exist in sufficient concentrations above the inversion layer in the nocturnal boundary layer to be mixed downward upon breakup of the nocturnal inversion and affect surface measurements. Instead, concentrations of NO and NOY were apparently mixed upward during the morning and midday hours by vertical boundary layer processes. Thus, the association of observed increases in surface NO and NOY mixing ratios based solely on downward mixing processes is not justified in all cases, and other sources and processes for these increases must be considered, particularly over rural areas.

Application of Fuzzy and Sliding Mode Algorithms for the Control of pH System

Control of pH neutralization system for strong acid and strong base is studied utilising fuzzy logic, sliding mode and hybrid (fuzzy-sliding mode) control techniques. Models from literature is used in the simulation studies. The sliding mode controller is designed by using boundary layer and adaptive terms. The controllers designed are tested for set-point tracking and for disturbance rejection conditions. The results obtained for the fuzzy controller and fuzzy-sliding mode controller are satisfactory with robust perfonnance.

The coupling of hyperbolic and elliptic boundary value problems with variable coefficients

We consider a one-dimensional coupled problem for elliptic second-order ODEs with natural transmission conditions. In one subinterval, the coefficient ϵ>0 of the second derivative tends to zero. Then the equation becomes there hyperbolic and the natural transmission conditions are not fulfilled anymore. The solution of the degenerate coupled problem with a flux transmission condition is corrected by an internal boundary layer term taking into account the viscosity ϵ. By using singular perturbation techniques, we show that the remainders in our first-order asymptotic expansion converge to zero uniformly. Our analysis provides an a posteriori correction procedure for the numerical treatment of exterior viscous compressible flow problems with coupled Navier–Stokes/Euler models. Copyright © 2000 John Wiley & Sons, Ltd.

The Influence of Lateral Boundary Conditions on the Asymptotics in Thin Elastic Plates

Here we investigate the limits and the boundary layers of the three-dimensional displacement in thin elastic plates as the thickness tends to zero in each of the eight main types of lateral boundary conditions on their edges: hard and soft clamped, hard and soft simple support, friction conditions, sliding edge, and free plates. Relying on construction algorithms [M. Dauge and I. Gruais, Asymptotic Anal., 13 (1996), pp. 167--197], we establish an asymptotics of the displacement combining inner and outer expansions. We describe the two first terms in the outer expansion: these are Kirchhoff--Love displacements satisfying prescribed boundary conditions that we exhibit. We also study the first boundary layer term: when the transverse component is clamped, it has generically nonzero transverse and normal components, whereas when the transverse component is free, the first boundary layer term is of bending type and has only its nonzero in-plane tangential component.

Finite horizon H/sub ∞/ state-feedback control of continuous-time systems with state delays

The finite horizon H/sub /spl infin// control of time-invariant linear systems with a finite number of point and distributed time delays is considered. The controller is obtained by solving coupled Riccati-type partial differential equations. The solutions to these equations and the resulting controllers are approximated by series expansions in powers of the largest delay. Unlike the infinite horizon case, these approximations possess both regular and boundary layer terms. The performance of the closed-loop system under a memoryless zero-approximation controller is analyzed.

Laser Ablation of Machined Wood Surfaces. 1. Effect on End-Grain Gluing of Pine (Pinus silvestris L.) and Spruce (Picea abies Karst.)

Summary This report describes two attempts to test the effect of removing the layer of damaged cells—the mechanical weak boundary layer. In both trials, the wood has been laser ablated using different types of lasers with different wavelengths. The goal has been to determine whether the glue joint strength is influenced by the mechanical weak boundary layer and to show how the laser wavelength affects the glue joint strength. The statistical evaluation of the results shows however no great differences between glue joints made of ablated and glue joints made of unablated surfaces. Nor are there any real differences between the different lasers and different wavelengths. This may be because there are indeed no differences between ablated and unablated surfaces, in terms of a mechanical weak boundary layer, but it may also be due to cracks which occurred in the wood specimens, and it is also possible that the glue itself was too weak. An ESEM (Environmental Scanning Electron Microscope)—analysis showed that fracture occurred primarily in the glueline and not in the boundary layer. In addition to the ESEM analysis and the statistical evaluation, a theoretical FEM (Finite Element Method)—analysis has been used to explain the crack initiation in the second trial.

A model of a homogenized cavity corresponding to a multinozzle droplet generator for continuous ink-jet printers

We modelize the behavior of a vibrating fluid cavity. Small nozzles are uniformly distributed in one direction on a side of the cavity. By means of asymptotic expansion in powers of the smallest dimension of the cavity, including boundary layer terms, we get the convergence of the solution of the three-dimensional problem, as well as the convergence of the solution of the “homogenized” three-dimensional problem towards the solution of the same two-dimensional problem. Numerical experiments have been carried out in order to illustrate the previous theoretical results. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 821–842, 1998

ASYMPTOTIC ESTIMATES FOR THE MULTI-DIMENSIONAL ELECTRO-DIFFUSION EQUATIONS

The drift-diffusion equations can be studied in the framework of singular perturbation analysis as a small parameter characterizing the device goes to zero. A formal asymptotic expansion, which includes internal (and eventually boundary) layer terms can be derived by standard techniques of asymptotic analysis. We present in this paper L2 estimates for the difference between the solutions of the full system and the first term of the expansion which are valid for multi-dimensional devices close to equilibrium. These estimates are based on a uniform monotone property of the equations with respect to the small parameter and on a L∞ bound for the gradient of solutions of equations under divergence form whose coefficients are bounded, and have derivatives in one direction which are bounded with respect to all variables.

Edge layers in thin elastic plates

This paper deals with the asymptotics of the displacement of a thin elastic 3D plate when it is submitted to various boundary conditions on its lateral face: namely, hard and soft clamped conditions, and hard support. Of particular interest is the influence of the edges of the plate where boundary conditions of different types meet. Relying on general results of [M. Dauge and I. Gruais, Asymptotics of arbitrary order for a thin elastic clamped plate. I: Optimal error estimates. Asymptotic Anal. 13 (1996) 167–197] and [M. Dauge and I. Gruais, Asymptotics of arbitrary order for a thin elastic clamped plate. II: Analysis of the boundary layer terms, Asymptotic Anal. (1996) to appear] for the hard clamped case, we see that the clamped plate (hard and soft) admit strong boundary layers, in which are concentrated the edge layers, while the hard supported plate has no edge layer and even no boundary layer at all in certain situations. We conclude with hints about corner layers, in the case when the mean surface of the plate itself is polygonal.

Higher order bending and membrane responses of thin linearly elastic plates

The limit behaviors of three-dimensional displacements in thin linearly elastic plates, as the half-thickness ε tends to zero, is now known for various lateral boundary conditions (see [1], [5]). In the generic case one obtains that the leading term of the asymptotic series u 0 + ge 1+ε 2u 2 +… of the scaled displacement is a Kirchhoff-Love field. In this Note we investigate the case where this leading term vanishes, giving the structure of the first non-vanishing term u k and an error estimate for its deviation from the scaled solution u(ε) multiplied by ε −k. There are essentially only three new cases (uncoupling in membrane and bending). Finally, in these situations a boundary layer term of the same order as the actual leading term appears in a generic way.

A Condensation--Evaporation Problem in Kinetic Theory

A linear Boltzmann model is used for studying a condensation{evaporation prob- lem in a bounded domain. First the time asymptotic limit is derived, which solves the associated stationary problem. Then the Milne problem is discussed for the boundary layer. Finally a fluid approximation is obtained in the small mean free path limit with initial and boundary layers of zeroth order. Introduction. The kinetic description of a rareed gas can be given through the Boltzmann equation for the density function f(t;x;v) of particles with velocity v at position x and time t. A coarser theory consists of describing the gas as a continuous fluid with local density (t;x), velocity u(t;x), and temperature T (t;x) satisfying the Euler or Navier{Stokes equations. In the limit of small mean free path, the fluid dynamic equations may be derived from the Boltzmann equation through either a Hilbert or Chapman{Enskog expansion; see, e.g., (2, 8, 9, 12). However, the fluid dynamic limits fail near shocks and for general indata near spatial or temporal boundaries. Among the many studies of the boundary layer structure let us mention the following. In (3), the steady nonlinear Boltzmann equation for a gas with zero bulk velocity between two plates at two dierent temperatures is solved for a small mean free path, using a Chapman{Enskog expansion between the two plates. Here the fluid part of the solution contains Fourier's law for heat conduction which can be made to satisfy dierent temperature values at the two plates. This is why the boundary layer terms only need to be of rst order with respect to the mean free path. An analogous study also including the initial layer is performed in (16) for the linear semiconductor case where further references in the eld may also be found. For more results in the area see also (5, 10, 13, 19). The present paper addresses the added presence of condensation{evaporation on the boundary. In this context a formal analysis and numerical computations are carried out in (17, 18) for a rareed gas with varying temperatures and condensation{ evaporation on the boundaries. On the basis of the linearized Boltzmann equation for hard sphere molecules, zeroth-order boundary layer terms are needed for solving the problem. Our paper considers the same problem for a rareed solute in a solvent gas, and with varying temperatures on the boundary. The linear Boltzmann equation is used as a model for the solute. We prove that a fluid approximation in the interior together with initial and boundary layer structures are available to describe the solute

A Priori Estimates for the Solution of Convection-Diffusion Problems and Interpolation on Shishkin Meshes

The solution of singularly perturbed convection-diffusion problems can be split into a regular and a singular part containing the boundary layer terms. In dimensions n = 1 and n = 2 , sharp estimates of the derivatives of both parts up to order 2 are given. The results are applied to estimate the interpolation error for the solution on Shishkin meshes for piecewise bilinear finite elements on rectangles and piecewise linear elements on triangles. Using the anisotropic interpolation theory it is proved that the interpolation problem on Shishkin meshes is quasi-optimal in L_{\infty} and in the energy norm.