On the unsteady three-dimensional boundary layer freely interacting with the external stream

Asymptotic equations that define unsteady processes in a three-dimensional boundary layer with self-induced pressure are derived. The pressure gradient under conditions of free interaction is, as usually, calculated not by the solution of the external problem of flow over a body, but on the assumption that it is due to growth of streamline displacement thickness near the body surface. Besides the principal terms, terms of second order of smallness are retained in asymptotic sequencies. If the characteristic dimensions of the free interaction region are the same in all directions in the plane tangent to the body surface, the system of equations defining the thin layer next to the wall must be integrated together with the system which defines the nonviscous stream.     

Investigation of the effect of the wall roughness and free stream turbulence on the boundary layer development

Development of a flat plate boundary layer (grad P = 0) was investigated in the closed type wind tunnel (0.5 × 0.9 × 2.7 m³). The plate was either aerodynamically smooth or covered with a thin plate (10 mm thick) with the surface made from the sand paper (grits 80). FST is generated by square mesh plane grids/screens set across the flow as to produce homogeneous, close to isotropy turbulence with various scales of velocity and length. The surface roughness was of transient type. The comparison is made of the effect of various grouping of the wall roughness and FST on the boundary layer development. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic theory of perturbations inducing a pressure gradient in a transonic flat-plate boundary layer

The role of asymptotic approaches to the study of viscous-inviscid interaction mechanisms in transonic outer flows is discussed. It is noted that there are several versions of multideck asymptotic constructions describing the self-induced pressure effect in transonic boundary layers. The asymptotic theory is used to uncover the internal structure of fluctuation fields, to treat instability-generating processes, and to analyze the behavioral features of linear and nonlinear wave fluctuations. Additionally, the properties of the eigenspectrum are described.     

The flow of pre-heated,two-dimensional laminar jet issuing into an aligned uniform stream

The aim of this paper is to establish the heat transfer characteristics of the flow downstream of a heated jet source of momentum issuing into an aligned uniform stream. Perturbation solutions about limiting similarity states at the jet and downstream are obtained. The formulation allows also for full numerical solution. A satisfactory reconciliation between numerical and perturbation solutions is demonstrated.     

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.     

On the boundary layer displacement effect near the trailing edge of an aft-loaded airfoil

Summary Viscous interaction effects near the trailing edge of an aft-loaded airfoil were investigated. A set of boundary layer data and the associated static pressure distributions are compared with predictions of computation methods, and reasons for the observed discrepancies are discussed.

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Zusammenfassung Die Wechselwirkung zwischen der Grenzschicht und des äusseren Strömungsfeldes in der Nähe der Hinterkante eines superkritischen Flügelprofils wird durch Vergleich rechnerischer Lösungen mit experimentell gewonnenen Ergebnissen untersucht, und die Gründe für die beobachteten Unstimmigkeiten erörtert.

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Dedicated to Professor Dr. Nicholas Rott, on the occasion of his 60th birthday, in appreciation of the inspiration received from him for the study of viscous interaction phenomena     

Receptivity of the boundary layer with separation bubble to external sound waves

The receptivity problem of laminar separated two-dimensional boundary-layer under influence of external acoustic waves is considered. Basic features of this theory were formulated by Ruban (1984) and then Goldstein (1985), but these investigations were limited by small perturbations of steady parallel flow only. The current paper takes into account essentially nonparallel regimes of base flow, including attached flows, flow with separated bubbles and the base flow at marginal value of parameter. The steady flow near a corner point of profile is of considerable interest and a question naturally arises about the sensitivity of separation bubble inside the boundary layer to external disturbances such as sound waves. Another question arises about the sensitivity of near critical base flow, namely if corner is concave there is a marginal angle which limits existence of such base flow. This fact usually is linked with sudden reconstruction of full flow pattern past a profile. Development of perturbations inside the interaction region may give some answers about the appearance of turbulence. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Steady boundary layer flow and reactive mass transfer past an exponentially stretching surface in an exponentially moving free stream

An analysis is made to study the steady two-dimensional boundary layer flow and reactive mass transfer past an exponentially stretching sheet in an exponentially moving free stream. The reaction rate of solute and the wall concentration distribution are taken variable. The governing equations are transformed and then solved numerically. The study reveals that the momentum boundary layer thickness is considerably smaller than that of stagnation point flow over stretching sheet. Due to increase of Schmidt number and reaction rate parameter the mass transfer considerably enhances. Importantly, for solute distribution, in addition to mass transfer, mass absorption occurs in certain situations.     

Free-convection boundary layer on an isothermal horizontal cylinder

The free convection on a horizontal circular cylinder whose temperature is suddenly increased is studied at large Grashof number. An accurate numerical method is described for determining the solution of the time-dependent boundary-layer equations. The development of the various physical properties of the flow are calculated and compared with their values at small and large times as obtained from the previously obtained analytical solutions.

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Résumé Cette étude considère la convection libre à nombre de Grashof élevé au dessus d'un cylindre horizontal de section circulaire lorsqu'on augmente brusquement sa température. On présente une méthode numérique qui permet de déterminer avec précision la solution des équations instationnaires de la couche limite. Le développement des caractéristiques physiques de l'écoulement est calculé et les résultats sont comparés avec ceux qui ont été déjà obtenus analytiquement pour les petites et grandes valeurs de la variable du temps.

     

The problem of the turbulent boundary layer

Summary Existing measurements of low-speed turbulent surface friction on a flat plate, in the absence of pressure gradient and roughness, are shown to be consistent with a simple analysis based on functional similarity in the velocity profile. In particular, the fully developed turbulent boundary layer is found to be unique within the accuracy of the experimental data, with uniqueness defined as the existence of a definite correspondence between local friction coefficient and momentum thickness Reynolds number. The relationships known as the law of the wall and the velocity defect law are found to describe the turbulent velocity profiles accurately for a considerable range of Reynolds numbers, and an effort is made to clarify the physical significance of these formulae. Finally, the proper definition of a length Reynolds number is discussed in terms of the asymptotic local properties of the ideal boundary layer, and numerical values for ideal mean and local friction coefficients are tabulated against Reynolds numbers based on momentum thickness and on distance from the leading edge.

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Zusammenfassung Es wird gezeigt, dass vorhandene Messungen der turbulenten Wandschubspannung an der glatten ebenen Platte in inkompressibler Strömung ohne Druckgradient durch eine einfache Berechnung in Übereinstimmung gebracht werden können. Die Rechnung beruht auf einer funktionellen Ähnlichkeit der Geschwindigkeitsverteilung. Es wird im besonderen gefunden, dass die vollentwickelte turbulente Grenzschicht innerhalb der Messgenauigkeit einem eindeutigen Zusammenhang zwischen dem örtlichen Reibungskoeffizienten und der Reynoldsschen Zahl, bezogen auf die Impulsdicke, folgt. Die Beziehungen, die als Wandgesetz und Mittengesetz bekannt sind, beschreiben die Geschwindigkeitsverteilung genau innerhalb eines erheblichen Bereiches Reynoldsscher Zahlen, und es wird versucht, den physikalischen Inhalt dieser Gesetzmässigkeiten zu vertiefen. Abschliessend wird eine zweckmässige Definition der auf Plattenlänge bezogenen Reynoldsschen Zahl diskutiert, die auf dem asymptotischen örtlichen Zustand der idealen Grenzschicht beruht. Rechenwerte der idealen, mittleren und örtlichen Reibungskoeffizienten, bezogen auf beide obigen Definitionen der Reynoldsschen Zahl, werden tabelliert.

     

Existence of solutions of laminar boundary layer equations with decelerating external flows

New results on the existence and nonexistence of solutions of laminar boundary layer equations with decelerating external flows are obtained. Some previous results treated these equations with accelerating or constant external flows. Our approach is to establish a system of two integral equations with singularities which will be proved to be equivalent to the laminar boundary layer equations and prove that the system has solutions by using the Leray–Schauder fixed point theorem and the Helly selection principle.     

The boundary layer on a rotating ellipse

Summary The boundary layer at the surface of an elliptic cylinder is considered, when the cylinder is set into uniform rotation about an axis through its centre, in a fluid that is otherwise at rest. It is shown, by numerical solution of the governing equations that, except for the limiting case of a circular cylinder, the solution develops a singularity at a finite time, which is interpreted as the onset of flow separation.

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Zusammenfassung Wir betrachten die Grenzschicht an der Oberfläche eines elliptischen Zylinders welcher in gleichmäige Drehung um sein Zentrum in einer sonst ruhenden Flüssigkeit versetzt wird. Es wirel durch rechnerische Lösung der bestimmenen Gleichunglen gezeigt, daß mit Ausnahme des Grenzfalles eines kreisförmigen Zylinders die Lösung eine Singularität nach endlicher Zeit annimrut welche als Begin der Flussseparation interpreties wired.

     

The boundary layer on a rotating sphere

Zusammenfassung Zur Berechnung der laminaren Grenzschicht auf einer rotierenden Kugel ist ein Differenzenverfahren entwickelt worden. Man kann sich dieses Verfahrens bedienen, um die Lösung schrittweise vom Pol zum Äquator zu erhalten. Die hier durchgeführten Berechnungen werden mit anderen Resultaten verglichen, welche man durch Reihenentwicklungen erhält.     

Homogenization of boundary layer of pseudo-plastic fluid in the presence of rapidly oscillating external forces

In this paper, we study the homogenization problem for equations of magnetohydrodynamic boundary layer of pseudo-plastic fluid. It is assumed that the external flow velocity and the external magnetic field are described by oscillating functions and the frequency depends on a small parameter. In von Mises variables and in Cartesian variables, we construct the homogenized problem, establish strong convergence of solutions in a special norm, and estimate the rate of this convergence. We show that in von Mises variables the convergence rates in different norms are of different orders of smallness.     

The transverse curvature effect on the incompressible laminar boundary layer for longitudinal flow over a cylinder

A solution to the partial differential equations governing incompressible laminar flow in the axial direction over a circular cylinder is presented. A method is employed which reduces the partial differential equations to a set of ordinary differential equations which are then solved consecutively. The solution is initiated at a leading edge and proceeds far downstream. Velocity profiles and values for the coefficient of skin friction, displacement thickness and momentum thickness are obtained. The results are compared to previously obtained solutions valid near the leading edge and asymptotic solutions valid far downstream.     

The effect of suction on boundary layer for rotating flows with or without magnetic field

The effect of suction on the steady laminar incompressible boundarylayer flow for a stationary infinite disc with or without magnetic field, when the fluid at a large distance from the surface of the disc undergoes a solid body rotation, has been studied. The governing coupled nonlinear equations have been solved numerically using the shooting method with least square convergence criterion. It has been found that suction tends to reduce the velocity overshoot and damp the oscillation.     

The laminar boundary layer on a blunted wedge

Zusammenfassung Es wird die Grenzschicht an einem abgestumpften Keil mit Hilfe der Görtlerschen Reihe berechnet. Man findet, von der Staupunkt-Grenzschicht ausgehend, eine asymptotische Annäherung an die Grenzschichten von Falkner und Skan; die asymptotische Lösung wird mit Hilfe der Reihe sehr befriedigend dargestellt, besonders nach Anwendung von verschiedenen Methoden zur Verbesserung der Konvergenz. Zudem wird die Annäherung an die Lösung im Unendlichen analytisch untersucht, wobei sogenannte asymptotische Eigenfunktionen auftreten.Die rechnerisch ausgewertete Lösung für einen Keil mit rechteckigem Öffnungswinkel kann als Vergleichsfall zur Beurteilung von numerischen Methoden benützt werden.

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To ProfessorHenry Görtler on his sixtieth birthday

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The research carried out at UCSD was supported by the Advanced Research Projects Agency (Project DEFENDER) under Contract No. DA-31-124-ARO-D-257, monitored by the U.S. Army Research Office-Durham.

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The research carried out at Stanford was supported by the Air Force Office of Scientific Research under Contract No. AF49(638)-1274.