Laminar boundary layer on a flat plate at high Prandtl number

Zusammenfassung Das Problem der Laminarströmung einer kompressiblen Flüssigkeit längs einer ebenen Platte bei grossen Prandtlzahlen wird mit Hilfe der Methode der inneren und äusseren Expansion gelöst unter der Annahme, dass das Produkt aus Dichte und Viskosität quer zur Grenzschicht konstant ist. Die Tabellen des Anhanges ermöglichen die Berechnung einer vollständigen Temperaturverteilung in der Grenzschicht für isolierte und nichtadiabatische Wände. Es erweist sich, dass die für den Rückgewinnungsfaktor gewonnene Formelr=1,922 ^1/3–1,341 eine wesentliche Verbesserung gegenüber früheren Ergebnissen ist, selbst bei =10³. Es wird eine Interpolationsformel vorgeschlagen, die in die exakte asymptotische Lösung bei grossem übergeht und beir=1 ebenfalls =1 ergibt.     

Estimation of the critical viscous sub-layer in shock wave boundary layer interaction

Zusammenfassung Wenn eine kompressible Grenzschicht (laminar oder turbulent) einer Störung unterworfen wird, sei es durch eigene Instabilität oder von aussen aufgeprägt, so macht sich der Einfluss dieser Störung sowohl stromaufwärts wie stromabwärts in der Strömungsgrenzschicht bemerkbar.Lighthill nimm in seiner mathematischen Behandlung des Mechanismus der Fortpflanzung dieser Störung stromaufwärts und stromabwärts an, dass die laminare Unterschicht der Grenzschicht genug ist, um Kompressibilitätseffekte darin vernachlässigen zu können.In vorliegender Veröffentlichung wurde die Dicke dieser kritischen laminaren Unterschicht abgeschätzt, ausgehend von ähnlichkeitsbetrachtungen und unter Verwendung vorhandener Messresultate. Es wurde festgestellt, dass das Verhältnis zwischen der kritischen, zähen Unterschicht zur totalen Grenzschtdicke (ungestört) im laminaren Fall von der Grössenordnung 110 und im turbulenten Fall etwa 1100 ist.     

Laminar boundary layer on rotating sphere and spheroids in non-Newtonian fluids

Zusammenfassung Die dreidimensionalen Grenzschichtgleichungen für nicht-Newtonsche Flüssigkeiten werden in orthogonalen krummlinigen Koordinaten angegeben. In Kapitel 3 wird die von der Umdrehung einer Kugel in einer sonst ungestörten nicht-Newtonschen Flüssigkeit innerhalb der laminaren Grenzschicht erzeugte Strömung untersucht. Die Kapitel 4 und 5 behandeln in analoger Weise das verlängerte und verkürzte Sphäroid. In allen drei Fällen zeigen die Lösungen einen Zustrom an den Polen, der von einem Abfluss am Äquator ausgeglichen wird.     

Laminar boundary layer over a flexible surface in oscillatory flow

A two-dimensional oscillatory flow over a flat flexible surface is analysed. Low and high frequency solutions are developed separately. Results depicting the effect of surface flexibility on the flow in comparison to that over a rigid surface are presented.     

Stability of laminar boundary layer along longitudinally curved walls

Summary To a former work [1] the perturbations are determined in the whole flow field. This also yields more exact eigenvalues.

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Zusammenfassung Zu einer früheren Arbeit [1] werden die Störungen im ganzen Strömungsbereich errechnet. Damit ergaben sich auch genauere Eigenwerte.

     

On the shear boundary layer of tension plates

The present paper deals with a generic class of problems for plates subjected to loadings combining a high in-plane tension and a small lateral pressure. It develops the governing differential equations in the singular pertubation form, through the postulation of retaining only one of the Kirchhoff's assumptions, that the plate thickness in the boundary layer region is invariant. The solution by using the standard perturbation method is discussed. The postulation is justified when it is demonstrated that in the shear boundary layer the plate thickness is of higher-order smallness. The general method of solution by the standard perturbation technique is applied to an annular plate problem. Problems of different combinations of supports at the inner and the outer boundaries are solved. The case in which both edges are simply supported is presented as an illustration of the solution technique. In other cases results only are presented. The effect of support on the boundaries is also discussed. The shear effect is found to be most significant at a clamped edge. In the special geometry, it is possible to demonstrate that, when the condition on membrane force is not met as required in the general theory, thagnitude of the boundary layer changes. Specifically, the paper presents a case in which the membrhich the membrane force is zero at the inner edge.     

Nonlinear evolution of a first mode oblique wave in a compressible boundary layer: I. Heated/cooled walls

The nonlinear stability of an oblique mode propagating in atwo-dimensional compressible boundary layer is considered underthe long wavelength approximation. The growth rate of the waveis assumed to be small so that the ideas of unsteady nonlinearcritical layers can be applied. It is shown that the spatial/temporalevolution of the mode is governed by a pair of coupled unsteadynonlinear equations for the disturbance vorticity and density.Expressions for the linear growth rate show clearly the effectsof wall heating and cooling, and in particular how heating destabilizesthe boundary layer for these long wavelength inviscid modesat O(1) Mach numbers. A generalized expression for the lineargrowth rate is obtained and is shown to compare very well fora range of frequencies and wave angles at moderate Mach numberswith full numerical solutions of the linear stability problem.The numerical solution of the nonlinear unsteady critical layerproblem using a novel method based on Fourier decompositionand Chebyshev collocation is discussed and some results arepresented.     

The behaviour of populations in a partially ionized flat plate boundary layer

Summary The behaviour of populations of the argon atoms in a partially ionized flat plate boundary layer has been investigated for the freestream conditionsT _e =3000 K,T _a =1000 K,N _e =10¹³ cm^–3,N _a =10¹⁶ cm^–3. The populations have been calculated from steady state conditions, and the net production rate of electron-ion and the electron energy transfer rate due to inelastic collisions have been expressed in terms of populations and rate constants for excitation-deexcitation and ionization-recombination. The result shows that the contributions of excitation-deexcitation to the electron energy transfer is quite large rather than those of ionization-recombination.

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Zusammenfassung Das Verhalten der Besetzungsdichten von Argon-Atomen in einer teilionisierten Grenzschicht längs einer flachen Platte ist für die folgenden Anströmbedingungen vonT _e =3000 K,T _a =1000 K,N _e =10¹³ cm^–3 undN _a =10¹⁶ cm^–3 untersucht worden. Die Besetzungsdichten sind unter der Annahme eines stationären Zustands berechnet worden; und die Nettoproduktionsrate von Elektron-Ion-Paaren und die Rate der auf unelastischen Stößen beruhenden Elektronenenergieübertragung ist mit Besetzungsdichten und Ratenkonstanten von Exzitations-Deexzitations- und Ionisations-Rekombinations-Prozessen ausgedrückt worden. Das Ergebnis zeigt, daß die Beiträge der Exzitations-Deexzitations-Prozesse zur Elektronenenergieübertragung viel größer sind als diejenigen von Ionisations-Rekombinations-Reaktionen.

     

Laminar compressible boundary layer flow at a three-dimensional stagnation point with vectored mass transfer

The effect of surface mass transfer velocities having normal, principal and transverse direction components (‘vectored’ suction and injection) on the steady, laminar, compressible boundary layer at a three-dimensional stagnation point has been investigated both for nodal and saddle points of attachment. The similarity solutions of the boundary layer equations were obtained numerically by the method of parametric differentiation. The principal and transverse direction surface mass transfer velocities significantly affect the skin friction (both in the principal and transverse directions) and the heat transfer. Also the inadequacy of assuming a linear viscosity-temperature relation at low-wall temperatures is shown.     

Interaction between a normal shock wave and a turbulent boundary layer at high transonic speeds. Part I: Pressure distribution

Summary Asymptotic solutions are derived for the pressure distribution in the interaction of a weak normal shock wave with a turbulent boundary layer. The undisturbed boundary layer is characterized by the law of the wall and the law of the wake for compressible flow. In the limiting case considered, for high transonic speeds, the sonic line is very close to the wall. Comparisons with experiment are shown, with corrections included for the effect of longitudinal wall curvature and for the boundary-layer displacement effect in a circular pipe.

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Zusammenfassung Asymptotische Lösungen für den Druckverlauf bei der Wechselwirkung zwischen einem schwachen normalen Stoss und einer turbulente Grenzschicht werden hergeleitet. Das Wandgesetz und Geschwindigkeitsdefekt-Gesetz für kompressible Strömung kennzeichnen die ungestörte Grenzschicht. Der Grenzfall hoher transsonischen Strömung, in dem die Schallinie in der Nähe der Wand liegt, wird untersucht. Die theoretischen Ergebnisse werden mit Experimenten verglichen. Dabei wird die Wandkrümmung und im Fall der Rohrströmung die Verdrängungsdicke berücksichtigt.

     

On the stress singularities and boundary layer in moderately thick functionally graded sectorial plates

In this paper, the stress analysis of moderately thick functionally graded sector plate is developed for studying the singularities in vicinity of the vertex and effects of boundary layer. Based on the first-order shear deformation plate theory, the governing partial differential equations are obtained. Using an analytical method, the stretching and bending equilibrium equations are decoupled. Also, introducing a function, called boundary layer function, the three bending equations are converted into two decoupled equations. These equations are solved analytically and the effects of boundary layer function on stress components are shown. Also, the singularities of shear force, moment resultants and boundary layer function are discussed for both salient (α?180)$(\alpha ?180)$ and re-entrant (α>180)$(\alpha >180)$ sectorial plates. In order to verify the accuracy of the results, the governing equations are also solved using differential quadrature method (DQM). By comparing the results of exact method with DQM, a good agreement can be seen.     

A note on the approximation of Dirichlet boundary control problems for the wave equation on curved domains

We consider an exact boundary control problem for the wave equation with given initial and terminal data and Dirichlet boundary control. The aim is to steer the state of the system that is defined on a given domain to a position of rest in finite time. The optimal control that is obtained as the solution of the problem depends on the data that define the problem, in particular on the domain. Often for the numerical solution of the control problem, this given domain is replaced by a polygon. This is the motivation to study the convergence of the optimal controls for the polygon to the optimal controls for the given domain. To study the convergence, the values of the optimal controls that are defined on the boundaries of the approximating polygons are mapped in the normal directions of the polygon to control functions defined on the boundary of the original domain. This map has already been used by Bramble and King, Deckelnick, Guenther and Hinze and by Casas and Sokolowski. Using this map, we can show the strong convergence of the transformed controls as the polygons approach the given domain. An essential tool to obtain the convergence is a regularization term in the objective functions to increase the regularity of the state.