Asymptotic solution of the unsteady boundary layer equations

Summary An extension of the Meksyn asymptotic method to unsteady boundary layers in laminar, incompressible flow is investigated. The results indicate that unsteady boundary layers can be calculated by the Meksyn asymptotic method with comparable accuracy to that obtained for steady flows. Several differences from the well developed steady-flow application exist and require further work before general problems can be treated. The calculation technique is more straight-forward for cases involving acceleration because three or four terms in the expansions may then yield sufficient accuracy. The form of the governing equation required by the Meksyn method indicates that it is most useful for unsteady stagnation boundary layers since some basic unsteady flows are not directly accessible in their simplest form from that equation. The effect of unsteadiness on the rate of asymptotic convergence is assessed by detailed comparison of a similar solution for unsteady, stagnation flow with analogous results from the Falkner-Skan equation and of reliable numerical results for both cases.

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Résumé On étudie une extension de la méthode asymptotique de Meksyn aux couches limites instables des écoulements laminaires de fluides incompressibles. Les résultats montrent que les couches limites instables peuvent être calculées à l'aide de la méthode asymptotique de Meksyn avec une précision comparable à celle obtenue pour les écoulements stables. Plusieurs différences existent par rapport à l'application, bien mise au point, aux écoulements stables; elles demandent encore du travail avant que les problèmes généraux puissent être traités. La méthode de calcul est plus directe dans les cas impliquant une accélération, car 3 ou 4 termes dans les développements assurent alors une précision suffisante. La forme de l'équation principale nécessaire à la méthode de Meksyn indique qu'elle est très utile pour les couches limites instables au repos; en effet, certains écoulements instables de base ne peuvent être atteints directement dans leur forme la plus simple à partir de cette équation. L'effet de l'instabilité sur la vitesse de convergence asymptotique est établi grâce à une comparaison détaillée d'une solution analogue pour un écoulement instable stagnant avec les résultats semblables obtenus par l'équation Falkner-Skan, et des résultats numériques sûrs obtenus dans les deux cas.

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This work was supported by the Energy Research and Development Administration.     

Investigation of the residual stresses in the carbamide polymer-glass boundary layer

The residual stresses at the carbamide polymer-glass interface have been investigated for the first time. It has been established that a high residual stress level exists in the polymer-glass boundary layer. Methods of reducing the residual microstress level are indicated.     

Numerical solution for the model of vibrating elastic membrane with moving boundary

In this work, we are interested in obtaining an approximated numerical solution for the model of vibrating elastic membranes with moving boundary. The model is an extension of Kirchhoff’s model, which takes into account the change of size during the vibration. We apply the finite element method with a finite difference method in time to obtain an approximated numerical solution. Some numerical experiments are presented to show the effect of moving boundary effects in vibrating elastic membranes.     

Finite deflection analysis of elastic plate by the boundary element method

An integral equation formulation for finite deflection analysis of thin elastic plates is presented, based on general nonlinear differential equations which are equivalent to the von Kármán equations and by virtue of generalized Green identities. Boundary element discretization is applied and a relaxation iterative approach is employed to solve the nonlinear plate bending problems. A number of numerical examples are given; the results of computation are compared with the analytical solutions and good agreement is observed. It appears that the approach developed in this paper is effective.     

The use of the method of boundary states to analyse an elastic medium with cavities and inclusions

The analytical method of boundary states is developed and theoretically substantiated. A corollary of the Weierstrass theorem is proved according to which a function that is harmonic in a bounded, simply connected domain can be approximated by a series of homogeneous harmonic polynomials. A basis of the space of functions that are harmonic outside any neighbourhood of a point is constructed. An algorithm is developed for filling the basis of the space of the states of a multicavity elastic body. The method is used to solve a series of problems of determining of the stress-strain state of an unbounded elastic medium containing spherical cavities or inclusions with different boundary conditions: the boundary of the cavity is free (the Southwell problem), constrained or under conditions of contact with a rigid core. The effect of the width of the intercavity layer on the stress concentration is analysed in a non-axisymmetric problem with two cavities. The form of the relation between the mean-square discrepancy in the boundary conditions of the solution obtained and the number of elements in the basis is indicative of the numerical convergence of the solution of this problem.     

Direct solution method for the equilibrium problem for elastic stents

We are interested in numerical methods for approximating vector‐valued functions on a metric graph. As a model problem, we formulate and analyze a numerical method for the solution of the stationary problem for the one‐dimensional elastic stent model. The approximation is built using the mixed finite element method. The discretization matrix is a symmetric saddle‐point matrix, and we discuss sparse direct methods for the fast and robust solution of the associated equilibrium system. The convergence of the numerical method is proven and the error estimate is obtained. Numerical examples confirm the theoretical estimates.     

The collocation method for the solution of boundary integral equations

This article is devoted to investigating the approximate solutions to a class of boundary integral equations over a closed, bounded and smooth surface found via the collocation method. This article provides sufficient conditions for the convergence of this method in the space of continuous functions.     

Solution of nonstationary boundary layer problems by fastest descent method

We consider the formation of a boundary layer above a semi-infinite plate that moves with velocity V_t8=At ⁿ in a viscous incompressible fluid. The problem is solved by Meksyn's asymptotic method. First, second, and third approximation equations are obtained. A numerical calculation is performed for various n, and especially interesting results are obtained for –0.5 < n < 0. The numerical solution of the third approximation equation is consistent with other known results.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 51–55, 1988.     

Existence of nonlinear boundary layer solution to the Boltzmann equation with physical boundary conditions

In this paper, the existence of boundary layer solutions to the Boltzmann equation with two physical boundary conditions for hard sphere model is considered. The boundary condition is first imposed on incoming particles of diffuse reflection type and the solution tends to a global Maxwellian in the far field. Similar to the problem with Dirichlet boundary condition studied in [S. Ukai, T. Yang, S.H. Yu, Nonlinear boundary layers of the Boltzmann equation: I. Existence, Comm. Math. Phys. 236 (3) (2003) 373-393], the existence of a solution is shown to depend on the Mach number of the far field Maxwellian, and there is an implicit solvability conditions yielding the co-dimensions of the boundary data. At last, the specular reflection boundary condition is considered and the similar conclusions are obtained.     

Asymptotic solution of the axisymmetric contact problem for an elastic layer of incompressible material

The solution of the axisymmetric contact problem for an elastic layer made of incompressible material and clamped along the base is constructed by regular and singular asymptotic methods.     

Error bounds for the solution of nonlinear two-point boundary value problems by Galerkin's method

Letu be the solution of the differential equationLu(x)=f(x, u(x)) forx(0,1) (with appropriate boundary conditions), whereL is an elliptic differential operator. Letû be the Galerkin approximation tou with polynomial spline trial functions. We obtain error bounds of the form , where 0jm andmk2m+q,p=2 orp=,h is the mesh size andq is a non negative integer depending on the splines being used.This research was supported in part by the Office of Naval Research under Contract N00014-69-A0200-1017.     

On solution of higher approximations to boundary layer flows

Résumé Pour trouver des approximations d'un ordre élevé dans les problèmes de couche limite, l'auteur propose une méthode qui unifie le approches de Meksyn et Görtler. Les valeurs initiales, pour les approximations successives de Görtler, sont evaluées à l'aide de la methode de Meksyn. Toutes les valeurs initiales sont obtaneues par différentiation d'une seule équation. Les résultats obtenus sont au moins équivalent à ceux obtenus par la méthode de Hsu, avec considérablement moins de calculs.     

Improvement of numerical solution of boundary layer problems by incorporation of asymptotic approximations

Summary A method for improvement of the numerical solution of differential equations by incorporation of asymptotic approximations is investigated for a class of singular perturbation problems.Uniform error estimates are derived for this method when implemented in known difference schemes and applied to linear second order O.D.E.'s. An improvement by a factor of ⁿ⁺¹ can be obtained (where is the small parameter andn is the order of the asymptotic approximation) for a small amount of extra work. Numerical experiments are presented.     

The stability of a transonic boundary layer on an elastic surface

The perturbations in the boundary layer over an elastic surface when there is non-stationary free viscous-inviscid interaction at transonic velocities are investigated using a modified three-deck model. The modification consists of retaining the term with the second derivative with respect to time (the singular term of the transonic expansion), which occurs in the model of the Lin–Reissner–Tsien equation when it is derived from the complete equations for the velocity potential. This enables the equations of the model to be improved so that they more accurately describe non-stationary and non-linear phenomena. It is shown that the modified model enables perturbations, ignored when using the classical three-deck model, to be taken into account. The compliance on the surface may lead to a reduction in the perturbation growth rate.     

Approximate solution of the transient free convection laminar boundary layer equations

Zusammenfassung Es wird eine angenäherte Lösung der Gleichungen der laminaren Grenzschicht in freier Konvektionsströmung an einer halbunendlichen senkrechten Platte angegeben. Obwohl bereits Ähnlichkeitsvariablen für die transiente Strömung dieses Typus angegeben wurden, ist bis heute kein numerisches Ergebnis bekannt. Daher wird im vorliegenden Aufsatz das System der nichtlinearen, gekoppelten partiellen Differentialgleichungen durch Approximation gelöst. Durch ein Integrationsverfahren kann das System der Differentialgleichungen mit drei unabhängigen Variablen (zwei Ortskoordinaten und Zeit) auf ein solches mit zwei unabhängigen Variablen zurückgeführt werden. Das so verbleibende Gleichungssystem wird auf zwei verschiedene Arten behandelt. Die eine wird als eine Methode der Integralrelationen bezeichnet, die andere als Linienmethode (ein Verfahren der Differenzenrechnung). Nach beiden Methoden werden für verschiedene Prandtl-Zahlen Geschwindigkeits- und Temperaturprofile berechnet und mit experimentellen Ergebnissen verglichen.

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Nomenclature a constant - a _i approximating function, see equation (24) - b constant - b _i approximating function, see equation (24) - d _i approximating functions, see equation (33) - e _i approximating functions, see equation (34) - f _i weighting functions - g _i weighting functions - g acceleration due to gravity - G Grashof number - h heat transfer coefficient - k thermal conductivity - L characteristic length - L(u) differential operator onu - Nu Nusselt number - P Prandtl number - q heat flux - t time - T temperature - T ambient temperature - u ₁ dimensionless characteristic velocity,Gr ^–1/2(U₁ L/) - U velocity in theX direction - U ₁ dimensional characteristic velocity,U ₁=U/f(Y/GD) - V velocity in theY direction - W _k weighting functions, see equation (5) - X distance along plate - x dimensionless distance along plate,X/L - x increment onx for method of lines - Y distance from plate - y dimensionless distance from plate,Y/L - thermal diffusivity - coefficient of thermal expansion - dimensional boundary layer thickness - dimensionless boundary layer thickness, /L - a function of - dimensionless length,Y/ - temperature excess, (T-T) - _w wall temperature difference, (T _w–T) - _k approximating functions, see equation (4) - kinematic viscosity - dimensionless time,Gr ^–1/2( _t /L ²) - approximating functions, see equation (4) Work performed under the auspices of the U.S. Atomic Energy Commission.     

On a fast method for computing the field of waves reflected by a thin elastic layer

A method for computing the field of waves reflected by a thin elastic layer is proposed. The method takes into account all kinds of exchange type reflections and is based on the summation of kinematic and dynamic analogues.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 152–162, 1989.