Optimal coordinates for three-dimensional incompressible laminar boundary layers

This paper summarizes the basic theory of optimal coordinates for the three-dimensional incompressible boundary layers. Though in essence the results obtained in this paper form an extension of Kaplun's work in two dimensions, the additional considerations due to three-dimensionality have extended the scope and applicability of the theory of optimal coordinates to more complicated situations.

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Résumé Cet article résume la théorie fondamentale des coordinées optimales pour les couches limites incompressibles à trois dimensions. Les résultats donnés dans cet article ne sont essentiellement que la continuation des recherches de Kaplun faites sur les couches limites à deux dimensions; mais les considérations supplémentaires dues à la tri-dimensionalité ont permis d'augmenter la portée et les applications de la théorie des coordinées optimales à des situations plus compliquées.

     

Turbulent boundary layers with large streamline curvature effects

It has been shown that turbulent flows are greatly affected by streamline curvature. In spite of this and the fact that curved shear flows are frequently encountered in engineering applications, the predictions of such flows are relatively less developed than the predictions of two-dimensional plane flows. Recently, various attempts were made by different investigators; however, their methods are only successful when the product of the boundary layer thickness to the local surface curvature |k| is 0.05. The present paper investigates the more general case where 0.1|k|0.5. Results show that the calculated boundary-layer characteristics for arbitrary free stream conditions are in good agreement with measurements.

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Zusammenfassung Es ist gezeigt worden, dass turbulente Strömungen durch eine stromlinienförmige Krümmung stark beeinflusst werden. Trotzdem und trotz der Tatsache, dass gekrümmte Randströmungen häufig bei technischen Anwendungen gefunden werden, sind die Vorhersagen solcher Strömungen verhältnismässig weniger entwickelt als die Vorhersagen zweidimensionaler ebener Strömungen. In letzter Zeit sind von verschiedenen Forschern Versuche in dieser Richtung unternommen worden; jedoch waren ihre Methoden nur dann erfolgreich, wenn das Verhältnis der Grenzschichtdicke zur bestimmten örtlichen Oberflächenkrümmung |k|0,05 ist. Die vorgelegte Arbeit untersucht den allgemeineren Fall, bei dem 0,1|k|0,5 ist. Die Resultate zeigen, dass sich die berechneten Grenzschichteigenschaften für beliebig freie Strömungsbedingungen in befriedigender Uebereinstimmung mit den Messungen befinden.

     

A note on three-dimensional boundary layers

Zusammenfassung Eine Vereinfachung der dreidimensionalen Grenzschichtgleichungen wird erreicht, falls die Stromlinien der Aussenströmung mit einer einfach-unendlichen Familie geodätischer Linien in der Oberfläche des Körpers, an dem sich die Grenzschicht bildet, fast zusammenfallen.     

Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows

We continue the work of Lopes Filho, Mazzucato and Nussenzveig Lopes [10] on the vanishing viscosity limit of circularly symmetric viscous flow in a disk with rotating boundary, shown there to converge to the inviscid limit in L ²-norm as long as the prescribed angular velocity α(t) of the boundary has bounded total variation. Here we establish convergence in stronger L ² and L ^p -Sobolev spaces, allow for more singular angular velocities α, and address the issue of analyzing the behavior of the boundary layer. This includes an analysis of concentration of vorticity in the vanishing viscosity limit. We also consider such flows on an annulus, whose two boundary components rotate independently. Supported in part by NSF grant DMS-0456861.     

Self-similar solution of unsteady compressible three-dimensional stagnation-point boundary layers

Summary The self-similar solution of the unsteady laminar compressible boundary-layer flow with variable properties at a three-dimensional stagnation point with mass transfer has been obtained when the free-stream velocity varies inversely as a linear function of time. The resulting ordinary differential equations have been solved numerically using an implicit finite-difference scheme. The results are found to be strongly dependent on the parameter characterizing the unsteadiness in the free-stream velocity. The velocity profiles show some features not encountered in steady flows.

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Zusammenfassung Ähnliche Lösungen der nichtstationären laminaren kompressiblen Grenzschichtströmung mit veränderlichen Stoffgrößen ist für einen dreidimensionalen Staupunkt mit Massen-Übertragung erhalten worden, für den Fall, daß die Anströmung invers zu einer linearen Funktion der Zeit variiert. Es entsteht eine gewöhnliche Differentialgleichung, die numerisch mit einer impliziten Differenzenmethode gelöst wurde. Die Ergebnisse hängen stark von dem Parameter ab, der die Nicht-Stationärität der Anströmung kennzeichnet. Die Geschwindigkeitsprofile zeigen einige Eigenschaften die in stationärer Strömung nicht auftreten.

     

Coefficients of multivalent symmetric functions of bounded boundary rotation

In this paper the sharp coefficient estimate problem for the classesC _p(β, m) andV _p(k,m) of multivalent close-to-convex functions of order β and multivalent functions of bounded boundary rotation of at mostkπ, whose functions are given bym-fold symmetric gap series, have been discussed respectively for β≥1?p/m>0 andk≥2(m/p). Moreover, it is shown that every function inV _p(k,m) arep-valent close-to-convex; hencep-valent; ifk<2 (1+m/p).     

Numerical computation of three-dimensional turbulent boundary layers for the infinite swept wing

A numerical scheme for turbulent incompressible boundary layer is proposed. The formulation of the problem assumes translation invariance in cartesian coordinates. An anisotropic eddy viscosity model is considered. Numerical experiments show good agreement with a streamline coordinate formulation on the one hand and experimental data on the other hand.

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Résumé Un schéma de discrétisation des équations de la couche limite turbulente et incompressible est proposé. La formulation du problème s'appuie sur l'invariance de translation en coordonnées cartésiennes. Un modèle de viscosité turbulente anisotrope est considéré. Les résultats montrent une bonne concordance avec, d'une part, la formulation du problème en coordonnées lignes de courant, et d'autre part, les mesures expérimentales.

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This work was supported by the Swiss National Science Foundation under Grant No. 2.855-0.85.     

Cubic spline interpolation of functions with high gradients in boundary layers

The cubic spline interpolation of grid functions with high-gradient regions is considered. Uniform meshes are proved to be inefficient for this purpose. In the case of widely applied piecewise uniform Shishkin meshes, asymptotically sharp two-sided error estimates are obtained in the class of functions with an exponential boundary layer. It is proved that the error estimates of traditional spline interpolation are not uniform with respect to a small parameter, and the error can increase indefinitely as the small parameter tends to zero, while the number of nodes N is fixed. A modified cubic interpolation spline is proposed, for which O((ln N/N)⁴) error estimates that are uniform with respect to the small parameter are obtained.     

On boundary layers

The concept of boundary layers, introduced by A. Volberg in [7], is generalized from subsets of the unit disk to subsets of general non-tangentially accessible (NTA) domains. Capacitary conditions of Wiener type series of both necessary and sufficient type for boundary layers are presented and the connection between boundary layers and minimally thin sets is studied.     

Stability of one‐dimensional boundary layers by using Green's functions

The aim of this paper is to investigate the stability of one‐dimensional boundary layers of parabolic systems as the viscosity goes to 0 in the noncharacteristic case and, more precisely, to prove that spectral stability implies linear and nonlinear stability of approximate solutions. In particular, we replace the smallness condition obtained by the energy method [10, 13] by a weaker spectral condition. © 2001 John Wiley & Sons, Inc.     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

Limit theorems for random symmetric functions

In this paper, based on theorems for limit distributions of empirical power processes for the i.i.d. case and for the case with independent triangular arrays of random variables, we prove limit theorems for U- and V-statistics determined by generalized polynomial kernel functions. We also show that under some natural conditions the limit distributions can be represented as functionals on the limit process of the normed empirical power process. We consider the one-sample case, as well as multi-sample cases. Dedicated to Professor V. M. Zolotarev on his sixty-fifth birthday. Supported by the Hungarian National Foundation for Scientific Research (grant No. T1666). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

Minimizing symmetric submodular functions

We describe a purely combinatorial algorithm which, given a submodular set functionf on a finite setV, finds a nontrivial subsetA ofV minimizingf[A] + f[V A]. This algorithm, an extension of the Nagamochi—Ibaraki minimum cut algorithm as simplified by Stoer and Wagner [M. Stoer, F. Wagner, A simple min cut algorithm, Proceedings of the European Symposium on Algorithms ESA '94, LNCS 855, Springer, Berlin, 1994, pp. 141–147] and by Frank [A. Frank, On the edge-connectivity algorithm of Nagamochi and Ibaraki, Laboratoire Artémis, IMAG, Université J. Fourier, Grenbole, 1994], minimizes any symmetric submodular function using O(|V|³) calls to a function value oracle. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.A preliminary version of this paper was presented at the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) in January 1995. This research was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.