A model of unsteady flow separation from a cylinder without boundary layer equations

Presented at the seminar of the Department of Computational Mathematics, May 26, 1983.     

On the curvature of compressible boundary layer flows near separation

Summary Oswatitsch's analytical expression for the slope of a viscous flow separation or reattachment streamline is shown to be consistent with detailed experimental data on a supersonic boundary layer flow past a compression corner. An extension of his analysis is then given which yields a comparable new relation for the streamlinecurvature just above separation in either laminar or turbulent two-dimensional compressible boundary layer flows. This result is applied to examine the possible occurrence of Görtler streamwise vortices due to such curvature.

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Zusammenfassung Es wird gezeigt, daß die analytische Beziehung von Oswatitsch für den Neigungswinkel einer zähflüssigen Strömungsablösung (oder das Wiederanlegen), in Übereinstimmung ist mit detaillierten Versuchsergebnissen an einer Überschall-Grenzschichtströmung längs einer Kompressionsecke. Darüber hinaus wird eine Erweiterung dieser Theorie angegeben, welche eine vergleichbare Beziehung für die Stromlinienkrümmung im Bereich über dem Ablösepunkt, sowohl für laminare als auch turbulente zweidimensionale Grenzschichtströmung gibt. Dieses Ergebnis wird zu einer Abschätzung möglichen Auftretens von Görtlerschen Wirbeln in Strömungsrichtung infolge dieser Krümmung verwendet.

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Based on work supported partially by the US Air Force Office of Scientific Research under Contract F49620-76-C-0013 and by the Alexander Humboldt Foundation.     

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)     

On a double boundary layer in a nonlinear boundary value problem

The question of the isotopy of a quasiconformal mapping and its special aspects in dimension greater than 2 are considered. It is shown that an arbitrary quasiconformal mapping of a ball has an isotopy to the identity map such that the coefficient of quasiconformality (dilatation) of the mapping varies continuously and monotonically. In contrast to the planar case, in dimension higher than 2 such an isotopy is not possible in an arbitrary domain. Examples showing specific features of the multidimensional case are given. In particular, they show that even when such an isotopy exists, it is not always possible to perform an isotopy so that the coefficient of quasiconformality approaches 1 monotonically at each point in the source domain.     

Integral boundary problem in a layer

Translated from Matematicheskie Zametki, Vol. 53, No. 6, pp. 122–129, June, 1993.     

Sequences of functionals with a boundary layer

Siberian Mathematical Journal -     

Singularly perturbed boundary value problems with a power-law boundary layer

We consider boundary value problems for second-order singularly perturbed equations whose solution has a power-law boundary layer that occurs because the degenerate equation has multiple roots.     

Mathematical model of a kinetic boundary layer

The two-dimensional (plane) problem of a hypersonic kinetic boundary layer developing on a thin body in the case of a homogeneous polyatomic gas flow with no dissociation or electron excitation is considered assuming that energy exchange between translational and internal molecular degrees of freedom is easy. (The approximation of a hypersonic kinetic boundary layer arises from the kinetic theory of gases and, within the thin-layer model, takes into account the strong nonequilibrium of the hypersonic flow with respect to translational and internal degrees of freedom of the gas particles.) A method is proposed for constructing the solution of the given kinetic problem in terms of a given solution of an equivalent well-studied classical Navier-Stokes hypersonic boundary layer problem (which is traditionally formulated on the basis of the Navier-Stokes equations).     

Wave motions in a spatial boundary layer

The propagation of perturbations in a boundary layer under conditions when the velocity of the approaching stream may be both subsonic and supersonic is considered. With regard to the initial flow in the boundary layer it is assumed that it is stationary and possesses a spatial character which is caused by the external pressure gradient and not by the curvature of the body around which the flow occurs (boundary layers of this kind are extensively used in experiments at the present time). The linearized equations describing waves of vanishingly small amplitude are studied in detail. An analysis of the dispersion relation which links the frequency of the free oscillations with the components of the wave vector reveals a number of special features which are only present in motions with a three-dimensional velocity field. In particular, it is established that the Cauchy problem for the system of linear equations is ill-posed.     

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.     

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.

     

Perturbation analysis of a magnetohydrodynamic boundary layer flow

This paper presents a perturbation analysis study of the flow of an electrically conducting power-law fluid in the presence of a uniform transverse magnetic field over a stretching sheet. The perturbation solutions for small and large values of the mixed convection parameter are explored. The asymptotic behavior of the solutions was examined for different values of the power-law index and the magnetic parameter.     

Non-steady periodic boundary layer

Zusammenfassung Es wird die nichtstationäre Grenzschichtbewegung an einem Rotationskörper untersucht und die Störungsgeschwindigkeit bestimmt. Ähnlich wie im ebenen Fall verschwindet diese in grösserer Entfernung nicht und ist stationär.     

Nonequilibrium jet boundary layer in a polyatomic gas

The two-dimensional nonequilibrium hypersonic free jet boundary layer gas flow in the near wake of a body is studied using a closed system of macroscopic equations obtained (as a thin-layer version) from moment equations of kinetic origin for a polyatomic single-component gas with internal degrees of freedom. (This model is can be used to study flows with strong violations of equilibrium with respect to translational and internal degrees of freedom.) The solution of the problem under study (i.e., the kinetic model of a nonequilibrium homogeneous polyatomic gas flow in a free jet boundary layer) is shown to be related to the known solution of the well-studied simpler problem of a Navier-Stokes free jet boundary layer, and a method based on this relation is proposed for solving the former problem. It is established that the gas flow velocity distribution along the separating streamline in the kinetic problem of a free jet boundary layer coincides with the distribution obtained by solving the Navier-Stokes version of the problem. It is found that allowance for the nonequilibrium nature of the flow with respect to the internal and translational degrees of freedom of a single-component polyatomic gas in a hypersonic free jet boundary layer has no effect on the base pressure and the wake angle.     

On vortex perturbations in a free-interacting boundary layer

Large-scale structures with an inviscid, non-linear subdomain (deck) on the bottom of a boundary layer in the case of subsonic and transonic free stream velocities are considered. A class of locally inviscid perturbations with an internal line of discontinuity of the tangential velocity, which leads to the appearance of a free term on the right-hand side of the Benjamin-Ono equations, is investigated. The shape of the above-mentioned line is sought and it is determined from the solution of a system of one-dimensional non-stationary equations in which, apart from the Benjamin-Ono equation, a kinematic condition and an equation for the inviscid deck close to the wall also occur. An example of a periodic, non-linear solution is constructed and amplitude constraints which ensure its realization are formulated.