Matched Asymptotic Expansions Research Articles

Introduction of non-uniformity through linearization of the system governing turbulent, mixing of a scalar quantity in a 2-d mixing layer

The introduction of mathematical non-uniformity in the formulation of the turbulent mixing of a scalar quantity (mass, temperature, etc.) for a 2-d, free shear flow using Goertler's [ZAMM 22 (1942) 244] perturbation argument is discussed. Approximate, i.e. thin shear layer self-similar forms for mass, momentum and the scalar quantity are derived, and then linearized using Goertler's method. Though successful for the mean velocity field, the regular expansion yields inconsistent solutions for the transport of a scalar. Sources of the non-uniformity are identified using appropriate numerical methods for both non-linear and linear formulations. A consistent result is obtained by rescaling the independent variable and equation system and identifying dominant behavior. The results of this corrected formulation are shown to be consistent with the relationships obtained by the author using an approximate matched asymptotic expansion procedure.

The evolution of reaction–diffusion waves in a class of scalar reaction–diffusion equations: algebraic decay rates

In this paper, we consider an initial-boundary value problem for the generalized Fisher–Kolmogorov–Petrovskii–Piscounov (FKPP) equation of order m>1, with the non-negative initial data being of O( x − α ) at large x (dimensionless distance), where α≥1/( m−1) (the corresponding problem when α<1/( m−1) having been considered in detail by Needham and Barnes [Nonlinearity 12 (1999) 41–58]. Using matched asymptotic expansions we are able to obtain the complete structure of the solution to the initial-boundary value problem for large t (time), which exhibits the formation of a permanent form reaction–diffusion travelling wave structure. This being in contrast to the case when α<1/( m−1), where the large t (time) structure exhibits the formation of an accelerating phase wave.

Dynamics of nonlinear resonant slow MHD waves in twisted flux tubes

Abstract. Nonlinear resonant magnetohydrodynamic (MHD) waves are studied in weakly dissipative isotropic plasmas in cylindrical geometry. This geometry is suitable and is needed when one intends to study resonant MHD waves in magnetic flux tubes (e.g. for sunspots, coronal loops, solar plumes, solar wind, the magnetosphere, etc.) The resonant behaviour of slow MHD waves is confined in a narrow dissipative layer. Using the method of simplified matched asymptotic expansions inside and outside of the narrow dissipative layer, we generalise the so-called connection formulae obtained in linear MHD for the Eulerian perturbation of the total pressure and for the normal component of the velocity. These connection formulae for resonant MHD waves across the dissipative layer play a similar role as the well-known Rankine-Hugoniot relations connecting solutions at both sides of MHD shock waves. The key results are the nonlinear connection formulae found in dissipative cylindrical MHD which are an important extension of their counterparts obtained in linear ideal MHD (Sakurai et al., 1991), linear dissipative MHD (Goossens et al., 1995; Erdélyi, 1997) and in nonlinear dissipative MHD derived in slab geometry (Ruderman et al., 1997). These generalised connection formulae enable us to connect solutions obtained at both sides of the dissipative layer without solving the MHD equations in the dissipative layer possibly saving a considerable amount of CPU-time when solving the full nonlinear resonant MHD problem.

Dilute suspension of high inertia particles in the turbulent flow near the wall

A steady flow of a very dilute suspension of solid particles in the turbulent flow near the wall is considered. The particle phase is described by the system of mean field mass, momentum, and turbulent Reynolds stress conservation equations incorporating the closure approximation for the third-order correlation of particle fluctuating velocity; this closure approximation, as well as the mean field equations, were derived earlier from the PDF (probability density function) kinetic equation governing the particle transport in the turbulent flow. In the turbulent flow near the wall, particles can be characterized as “high inertia” or “low inertia” by small or large values, respectively, of a single nondimensional parameter: the normalized particle relaxation time combining the wall friction velocity and physical parameters of the fluid and particle phase. Making use of the technique of matching asymptotic expansions, the distributions are found of the normal and shear turbulent stress, volume fraction, and mean velocity in the particle phase of the suspension of high inertia particles.

논문 : 모따기 된 전향계단에 부딪치는 와류에 의한 유동소음

In cavity flow, the rectangular step generates so strong sound that many researchers have investigated method to suppress the nois during interaction between vortical flow and rectangular forward step. In this study the flow noise from the vortex motion in two-dimentional low Mach number flow past a forward step with various chamfering angle is calculated numerically. Inviscid incompressible discrete vortex model and matched asymptotic expansion(MAE) theory are applied to obtain the inner flow field and the outer noise field. Both source acoustic pressure and sound intensity are obtained with various chamfering height, chamfering angle and initial vortex position. The pressure amplitude is most suppressed when the chamfering angle is between and at the chamfering length of 30% of the step height.

Explicit Guidance of Aeroassisted Orbital Transfer Using Matched Asymptotic Expansions

An explicit guidance of aeroassisted orbital transfer by using the method of matched asymptotic expansions is presented. The aeroassisted orbit transfer comprises three consecutive phases: a descending phase entering atmospherewithanegativee ightpathangle,atransatmospherice ighttodepleteitsexcessenergy,andanascending e ight targeting the desired apogee. Feedback control via lift modulation is used to guide the vehicle during the atmospheric e y-through, whereas a matched asymptotic solution for the exit trajectory is available to aim the apogee of the target orbit, thus determining the pull-up condition. The feedback control ensures the stability of a trajectory around the nominal trajectory by compensating for the nonlinear terms in the motion of the vehicle. When the method of matched asymptotic expansions is used, the control algorithms for the explicit guidance are derived and tested against the 1976 U.S. Standard Atmosphere. Simulation results indicate that the control algorithms can effectively control the trajectories in both the lower atmosphere and in the exit phase under the targeting dispersions of atmospheric variations.

Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames

This Note is devoted to the justification of an asymptotic model for quasisteady three-dimensional spherical flames proposed by G. Joulin [7]. The paper [7] derives, by means of a three-scale matched asymptotic expansion, starting from the classical thermo-diffusive model with high activation energies, an integro-differential equation for the flame radius. In the derivation, it is essential for the Lewis number – i.e., the ratio between thermal and molecular diffusion – to be strictly less than unity. In this Note, we give the main ideas of a rigorous proof of the validity of this model, under the additional restriction that the Lewis number is close to 1. To cite this article: C. Lederman et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 569–574.

Quasi-Neutral Limit for Euler-Poisson System

This paper provides a rigorous proof of the quasi-neutral limit for the Euler-Poisson system on a bounded domain in one space dimension. The most general case is being considered when the plasma is sustained by ionization. A wide range of plasmas, from collisionless to highly collisional, is permitted. At the plasma center, the ions are assumed to be at rest, and essentially quasi-neutral initial data are prescribed. The theorem asserts that the quasi-neutral limit is obtained until the ion velocity reaches the ion-sound speed. In addition, formal matched asymptotic expansions are given which describe the solution in its passage from the plasma center to the wall.

Crack front rotation and segmentation in mixed mode I + III or I + II + III. Part I: Calculation of stress intensity factors

This paper and its companion are devoted to the theoretical analysis, compared with experiments, of crack front rotation and segmentation in brittle solids under mixed mode I + III or I +II + III loadings. A first, indispensable task, which is the topic of the present paper, consists of evaluating the stress intensity factors along the front of a crack after rotation or segmentation of this front. The method of analysis combines three elements. The first one is 3D matched asymptotic expansions. The second one is an analogy with the related problem of a planar crack, but with a slightly wavy front. The output of these two ingredients is the reduction of the search for the quantities of interest to the solution of some 2D, plane strain or antiplane elasticity problems. The third element consists of the use of Muskhelishvili's complex potentials formalism and conformal mapping to solve these problems.

Approximate factorization of Wiener–Hopf kernels

The key step in the solution of a Wiener–Hopf equation is the factorization of the Fourier transform of the kernel, K(α) (here, α is the Fourier variable). Exact factorizations are seldom useful, either because they do not exist or because they are too cumbersome to evaluate accurately. It is desirable, therefore, to construct approximations of K(α) which are both easy to factorize and simple to evaluate. Recently, Crighton examined the case that K depends on a small parameter, ε; i.e., K=K(α;ε). Crighton showed that matched asymptotic expansions (MAE) can be used to construct a composite asymptotic approximation to K for ε→0, valid uniformly over the full required range of α. The composite approximation comprises several relatively simple factors in typical applications, and these factors are then decomposed into their Wiener–Hopf factors by standard techniques. Abrahams has introduced an alternative approximation method for Wiener–Hopf kernels based on Padé approximants. Padé approximants give explicit uniform approximations to K valid in the full relevant range of α, do not require a small expansion parameter, and can be computed with relatively little effort. Although Crighton’s and Abrahams’s approaches seem at first glance unrelated, this talk will show that under certain circumstances, they are closely related.

Sound diffraction by an underwater ridge with soft finite impedance

Laboratory experiments have been conducted in a water tank to investigate sound propagation over an underwater ridge with soft, finite impedance, very low density (0.12 kg/m3) and sound speed (250 m/s), and very high attenuation (2–3 dB/cm in the frequency range of interest of 50–85 kHz). The diffracted acoustic pressure was measured along the ridge surface and along a vertical axis behind the ridge. Good agreement was found between predictions from the theory of matched asymptotic expansions (MAE) and the experimental data. The diffracted sound field is quite sensitive to the acoustic attenuation of diffraction material, thus showing potential as an inversion technique to characterize soft material attenuation.

Force suiveuse critique sur une colonne pesante semi-infinie : modèle et expériences

Using matched asymptotic expansions, we give here an approximate value of the follower force that results in the instability of a semi-infinite hanging beam. An experimental verification is shown in the context of fluid-conveying pipes. The transition between the classical short length solution and the proposed solution for infinite length is also observed in the experiments.

On the scattering of [formula omitted]-polarized electromagnetic field by an ideally conductive cylindrical body of a trapping type

We consider a two-dimensional analog of Helmholtz resonator with walls of finite thickness in the critical case when there exists an eigenfrequency which is the limit of poles generated by both the bounded component of the resonator and the narrow connecting channel. Under the assumption that the limit eigenfrequency is a simple eigenfrequency of the bounded component, the asymptotics of two poles converging to this eigenfrequency are constructed by using the method of matching asymptotic expansions. Explicit formulas for the leading terms of the asymptotics of poles and of the solution of the scattering problem are obtained.

Asymptotics of bound states and bands for waveguides coupled through small windows

A system of two waveguides coupled laterally through small windows is considered. The asymptotics (in the width of windows) of ground state close to the threshold is obtained for the case of finite number of apertures. The cases of periodic system of coupling windows is studied. The asymptotics of the band edges is obtained. The technique is matching of asymptotic expansions of the solutions.

Asymptotic factorization of Wiener–Hopf kernels

The key step in the Wiener–Hopf technique involves the factorization of a kernel function K( α), analytic in a strip D, into a product K +( α) K −( α), where K ±( α) are analytic in upper and lower halves R ± of the complex α-plane overlapping in the strip D, are free of zeros in R ± and have algebraic behaviour at infinity therein. Attention here is restricted to scalar kernels, for which explicit representations of the factors K ±( α) can be given in terms of Cauchy integrals. In many cases of interest, however, K≡ K( α, ϵ), where ϵ is a parameter taking small values. The issue is how to obtain useful approximate expressions for the factors K ±( α, ϵ) holding in all parts of R ± of interest, as ϵ→0. Limited exact results are known, due to Kranzer and Radlow, but the general issue remains open. It is proposed here that the method of matched asymptotic expansions (MAE) provides a general and essentially optimal approach to this issue. The strip D is divided into the relevant number of asymptotically distinct but overlapping domains, and in each of these K( α, ϵ) is appropriately approximated to leading order as ϵ→0. Then a multiplicative composite is formed, approximating K( α, ϵ) uniformly in D. This composite comprises several relatively simple factors in typical applications, and these factors are then decomposed into their WH factors by standard WH techniques such as the use of infinite products and Cauchy integrals. The power of this approach is demonstrated by formal applications to a number of WH kernels arising in wave theory problems. Explicit asymptotic formulae are derived, even for quite complicated kernels K( α, ϵ), in the asymptotic limit ϵ→0. The formal steps suggest ways in which, in any particular case, the process advocated could be made rigorous. It is also shown, for two particular but broadly illustrative examples, how higher order corrections can be completely calculated by MAE; the (limited) results of Kranzer and Radlow are recovered, and are supplemented by the construction of equally important terms not provided by Kranzer and Radlow, including expressions for the WH factors valid at infinity.

High Péclet number heat exchange between cocurrent streams

This paper concentrates on the analysis of the heat transfer between two cocurrent laminar flows in parallel channels. For high values of the Peclet number Pe, a boundary layer arises near the wall separating the streams. Matched asymptotic expansions (MAE) are used to obtain approximate solutions. We consider arbitrary inlet temperatures and derive higher-order corrections of the boundary problem. The separating wall is supposed to be sufficiently thin to neglect the heat conduction in it. Analyticity and adiabatic conditions at the outer walls impose restrictions on the inlet temperatures. It turns out, however, that only the inlet temperatures at the wall separating the two fluids enter the leading-order problem. The Nusselt numbers thus calculated are in the leading order proportional to (Pe/x)1/3, where x is the stream-wise coordinate. An estimate of the thickness of the separating wall to validate the MAE approach is obtained. It is also demonstrated that the MAE analysis is unable to describe the heat exchange of counterflowing fluids.

Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity

A large-Reynolds-number asymptotic solution of the Navier–Stokes equations is sought for the motion of an axisymmetric vortex ring of small cross-section embedded in a viscous incompressible fluid. In order to take account of the influence of elliptical deformation of the core due to the self-induced strain, the method of matched of matched asymptotic expansions is extended to a higher order in a small parameter ε = (v/Γ)1/2, where v is the kinematic viscosity of fluid and Γ is the circulation. Alternatively, ε is regarded as a measure of the ratio of the core radius to the ring radius, and our scheme is applicable also to the steady inviscid dynamics.We establish a general formula for the translation speed of the ring valid up to third order in ε. This is a natural extension of Fraenkel–Saffman's first-order formula, and reduces, if specialized to a particular distribution of vorticity in an inviscid fluid, to Dyson's third-order formula. Moreover, it is demonstrated, for a ring starting from an infinitely thin circular loop of radius R0, that viscosity acts, at third order, to expand the circles of stagnation points of radii Rs(t) and R˜s(t) relative to the laboratory frame and a comoving frame respectively, and that of peak vorticity of radius R˜p(t) as Rs ≈ R0 + [2 log(4R0/√vt) + 1.4743424] vt/R0, R˜s ≈ R0 + 2.5902739 vt/R0, and Rp ≈ R0 + 4.5902739 vt/R0. The growth of the radial centroid of vorticity, linear in time, is also deduced. The results are compatible with the experimental results of Sallet &amp; Widmayer (1974) and Weigand &amp; Gharib (1997).The procedure of pursuing the higher-order asymptotics provides a clear picture of the dynamics of a curved vortex tube; a vortex ring may be locally regarded as a line of dipoles along the core centreline, with their axes in the propagating direction, subjected to the self-induced flow field. The strength of the dipole depends not only on the curvature but also on the location of the core centre, and therefore should be specified at the initial instant. This specification removes an indeterminacy of the first-order theory. We derive a new asymptotic development of the Biot-Savart law for an arbitrary distribution of vorticity, which makes the non-local induction velocity from the dipoles calculable at third order.

Decay rates of internal waves in a fluid near the liquid-vapor critical point

We study the damping of internal waves in a viscous fluid near the liquid-vapor critical point. Such a fluid becomes strongly stratified by gravity due to its large compressibility. Using the variable-density incompressible Navier-Stokes equations, we model an infinite fluid layer with rigid horizontal boundaries and periodic side boundary conditions. We present operator-theoretic results that predict the existence of internal-wave modes with arbitrarily small damping rates. We also solve the eigenvalue problem numerically using a compound matrix shooting method and a second method based on a matched-asymptotic perturbation expansion for small viscosity. At temperatures far above the critical point, the damping of the internal waves is substantially influenced by both boundary layer and volumetric effects. The boundary layer effect is caused by horizontal shearing layers near the two fixed horizontal boundaries. As the temperature approaches the critical temperature, an additional internal shearing layer develops as the density stratification curve steepens on approach to the two-phase regime. Numerical calculations show that for some of the internal-wave modes this causes a dramatic increase in the damping rate that dominates the boundary layer effects.

Compact solutions for the corner singularity in bonded lap joints

This paper presents compact solutions of the corner singularity at the adhesive/adherend interface in a bonded lap joint. Two configurations of special importance to evaluating joint strength are considered: corners pertaining, respectively, to a square edge and a spew fillet. The stress intensity factors are determined for the limiting case of rigid substrates by means of a numerical matched asymptotic expansion method. The non-dimensional stress intensity factor depends only on the Poisson's ratio of the adhesive, and this variation is characterised in a convenient analytical form by polynomial expressions. Comparison with finite-element results of a bonded joint obtained using fine mesh near the corner points confirms that the present analytical solutions provide very good representations of the singular stress fields at adhesive/substrate corners.

Numerical experiments on the behavior of Rossby waves in the critical layer

The barotropic vorticity equation is numerically integrated in time to study the behavior of Rossby waves in the vicinity of the critical latitude. The integrations are conducted with linear, quasi-linear (wave-mean flow interaction), and nonlinear models on a sphere for both the inviscid and viscous cases. The meridional resolution required for the correct representation of a vorticity overturning is determined for both the inviscid and viscous cases. The Stewartson–Warn–Warn (SWW) analytical solution is reexamined by comparison with the present numerical results. The cat's eye represented by the closed streamfunction is tilted in the present model results, unlike that in a ‘scaled model’ based on the SWW solution. It is shown that this discrepancy is due to the omission of the meridional structure of the disturbance in the first-order matched asymptotic expansions in the SWW solutions.