The effects of suction on the nonlinear stability of a three-dimensional compressible boundary layer

Permanent address (for correspondence), School of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, UK. This study follows on from work by Bassom & Seddougui (1992,Proc. R. Soc. Lond. A 436, 405–15) on the effects of suctionon the nonlinear stability of the three-dimensional incompressibleboundary layer induced by a rotating disc. This flow has twotypes of stationary instability, one corresponds to the upper-branchinviscid mode and the other to the lower-branch viscous mode.This latter instability is characterized by an effective velocityprofile which has a zero shear stress at the wall and, as inBassom & Seddougui (1992), it is on this mode that interestis focused here. The effect of suction on the compressible flow,and its subsequent instability, is shown to be significant.     

The compressible Gortler problem in two-dimensional boundary layers

In this paper the authors investigate the growth rates of Görtlervortices in a compressible flow in the inviscid limit of largeGörtler number. Numerical solutions are obtained for O(1)wavenumbers. The further limits of (i) large Mach number and(ii) large wavenumber with O(1) Mach number are considered.It is shown that two different types of disturbance mode canappear in this problem. The first is a wall layer mode, so namedas it has its eigenfunctions trapped in a thin layer near thewall. The other mode investigated is confined to a thin layeraway from the wall and termed a trapped-layer mode for largewavenumbers and an adjustment-layer mode for large Mach numbers,since then this mode has its eigenfunctions concentrated inthe temperature adjustment layer. It is possible to investigatethe near crossing of the modes which occurs in each of the limitsmentioned. The inviscid limit does not predict a fastest growingmode, but does enable a most dangerous mode to be identifiedfor O(1) Mach number. For hypersonic flow the most dangerousmode depends on the size of the Görtler number.     

Tollmien-Schlichting/vortex interactions in compressible boundary-layer flows

A weakly nonlinear interaction of oblique Tollmien-Schlichtingwaves and longitudinal vortices in compressible, high Reynoldsnumber, boundary-layer flow over a flat plate is consideredfor all ranges of the Mach number. The interaction equationsconsist of equations for the vortex which is indirectly forcedby the waves via a boundary condition, whereas a vortex termappears in the amplitude equation for the wave pressure. Thedownstream solution properties of interaction equations arefound to depend on the sign of an interaction coefficient. Thisparticular type of weakly nonlinear interaction was first proposedby Hall & Smith (1989), who considered incompressible flows;however, there are some errors in their formulation. Correctedresults for the incompressible regime are presented for comparisonwith those calculated for compressible flows. Compressibilityis found to have a significant effect on the interaction properties,principally through its impact on the waves and their governingmechanism, the ‘triple-deck’ structure. It is foundthat, in general, the flow quantities will grow slowly withincreasing downstream coordinate. However, for flows with Machnumber values below 2, there exists a small band of wave obliquenessangles for which the solutions terminate in abrupt, finite-distance‘break-ups’.     

On the Nonlinear Development of Görtler Vortices in a Compressible Boundary Layer

The nonlinear development of the Görtler instability in compressible boundary layers on curved walls is considered for vortices of asymptotically large wavenumber. The starting point for our calculations lies in the work of Hall and Lakin (Proc. Roy. Soc. London Ser. A 415:421–444), where the incompressible results were formulated. Without neglecting downstream partial derivatives, the initial development of a vortex from the point where it first starts to grow is calculated. It is shown how the same basic structure that occurs in incompressible flow exists, where the disturbance is confined to a core region bounded above and below by thin shear layers, but that the flow in the core region is of more complicated form than that for incompressible flow.     

On the Non-linear Evolution of Gortler Vortices in Non-parallel Boundary Layers

The non-linear development of finite amplitude Görtlervortices in a non-parallel boundary layer on a curved wall isinvestigated using perturbation methods based on the smallnessof e, the non-dimensional wavelength of the vortices. The crucialstage in the growth or decay of the vortices takes place inan interior viscous layer of thickness O(²) and length O().In this region the downstream velocity component of the perturbationcontains a mean flow correction of the same order of magnitudeas the fundamental which is driving it. Moreover, these functionssatisfy a pair of coupled non-linear partial differential equationswhich must be solved subject to some initial conditions imposedat a given downstream location. It is found that, dependingon whether the boundary layer is more or less unstable downstreamof this location, the initial disturbance either grows intoa finite amplitude Görtler vortex or decays to zero. Forthe Blasius boundary layer on a concave wall it is found thatGörtler vortices can only develop if the rate of increaseof curvature of the wall is sufficiently large. In this casethe finite amplitude solution which develops initially in an-neighbourhood of the position where the disturbance is introducedchanges its structure further downstream. This structure isinvestigated at a distance O() (with 0< <1) downstreamof the above -neighbourhood. In this régime the downstreamfundamental velocity component has an elliptical profile overmost of the flow field. However, in two thin boundary layerslocated symmetrically either side of the centre of the viscouslayer the fundamental velocity component decays exponentiallyto zero. The locations of these layers are determined by aneigenvalue problem associated with the one-dimensional diffusionequation. The mean flow correction persists both sides of theboundary layer and ultimately decays exponentially to zero. This large amplitude motion is not sensitive to the imposedinitial conditions and appears to be the ultimate state of anyinitial disturbance. However, in the initial stages of the growthof the vortex, some surprising flows are possible. For example,it is possible to set up a vortex flow similar to that observedby Wortmann (1969) which consists of a sequence of cells inclinedat an angle to the vertical.     

Zur Theorie der dreidimensionalen Instabilität laminarer Grenzschichten

Summary In the case of large wave lengths a refined treatment of the three-dimensional instability of laminar boundary layers is made possible by taking into account a further term of the differential equations of the instability. The improved curve of the Görtler-ParameterRe (/R)^1/2 for neutral disturbances has a minimum with an instability wave length approximately equal to 19 , the boundary layer thickness. The results of [2] are not changed essentially, especially the critical values of the Görtler parameter are confirmed and also the fact, that the critical wave length is large.

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Diese Mitteilung ist im wesentlichen während der Arbeit des Verfassers am Mathematischen Institut der Universität Freiburg i. Br. entstanden.     

The Finite Dean Problem: Nonlinear Theory

The Dean problem models flows in curved pipes of square cross-sectionby considering pressure-driven axisymmetric flow between concentriccylinders. The inclusion of ends on these cylinders introducesa range of flows not seen in an infinite formulation. Weaklynonlinear analyses are made possible by considering end conditionssuch as those of Blennerhassett & Hall (1979). This paperextends to Dean flow this nonlinear work of Hall (1980). A newderivation of the amplitude equations will be given, which alsouses a perturbation from the intersection points of the oddand even curves of critical Taylor number against cylinder heightgiven by linear theory. Equilibrium solutions of the amplitudeequations are then found and their stability examined. Unlikethe Taylor-vortex results of Hall, the properties of these solutionsdepend on the particular intersection point under consideration.A comparison is given with previous work on the finite Deanproblem.     

The （G＇/G, 1/G）-expansion method and its application to travelling wave solutions of the Zakharov equations

The （G＇/G, 1/G）-expansion method for finding exact travelling wave solutions of nonlinear evolution equations, which can be thought of as an extension of the （G＇/G）-expansion method proposed recently, is presented. By using this method abundant travelling wave so- lutions with arbitrary parameters of the Zakharov equations are successfully obtained. When the parameters are replaced by special values, the well-known solitary wave solutions of the equations are rediscovered from the travelling waves.     

Further Studies of the Application of Constrained Minimization Methods to Fredholm Integral Equations of the First Kind

Permanent address: Department of Mathematics, University of Queensland, Australia. Following earlier work of Babolian & Delves (J. Inst. MathsApplics (1979) 24, 157–174) the Galerkin equations forintegral equations of the first kind are stablized by imposingasympotic decay rates on the expansion coefficients. Results for the formulation in the l₂ norm are compared withresults of Babolian & Delves where the l₁ norm was used. The importance of the choice of the constants which specifythe decay rates is also considered. Theoretical results andcomputational experiments show that previously used automaticselection of these constants needs to be safeguarded by monitoringthe residuals of the Galerkin equations.     

Second-order thermal boundary-layer on a blunted wedge

Summary Second-order thermal boundary-layer solutions are obtained for flow past a blunted wedge with constant wall temperature. Contributions due to longitudinal curvature and displacement effect are obtained by employing the Görtler power series method. The first five terms of the series for each of the effects are computed. Since the region of validity of the results thus obtained is restricted in the streamwise direction, Eulerization technique is used to extend the radius of convergence to infinity.

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Zusammenfassung Die thermische Grenzschichtlösung zweiter Ordnung wurde für die Strömung an einem abgestumpften Keil mit konstanter Temperatur bestimmt. Die Beiträge wegen Krümmung und Verdrängungsdicke wurden erhalten; die Methode der Görtler'schen Reihe wurde verwendet. Für jeden Effekt wurden die fünf ersten Koeffizienten berechnet. Da die Gültigkeit der Ergebnisse für grosse Werte der Variablen in Strömungsrichtung begrenzt ist, wurde die Technik der Euler'schen Konvergenzverbesserung verwendet, um den Konvergenzradius bis ins Unendliche zu erstrecken.

     

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.     

Neighborhood Complexes of Stable Kneser Graphs

It is shown that the neighborhood complexes of a family of vertex critical subgraphs of Kneser graphs—the stable Kneser graphs introduced by L. Schrijver—are spheres up to homotopy. Furthermore, it is shown that the neighborhood complexes of a subclass of the stable Kneser graphs contain the boundaries of associahedra (simplicial complexes encoding triangulations of a polygon) as a strong deformation retract.* The first author was partially supported by the G&ouml;ran Gustafsson Foundation for Research in Natural Sciences and Medicine. The second author was supported by the graduate school Algorithmische Diskrete Mathematik, which is funded by the Deutsche Forschungsgemeinschaft, grant GRK 219/3. The DAAD partially supported a stay at KTH, Stockholm, in December 1998, where this work was done: DAAD program AZ 313/S-PPP     

The Existence of Görtler Vortices in Separated Boundary Layers

The linear stability properties of Görtler vortices within a general separated boundary layer flow are addressed. There has been little previous theoretical work directed toward this problem and here we are able to characterize the important features of vortices over the complete wavenumber spectrum. This investigation complements earlier studies of vortices within an attached flow which demonstrated that there are three distinctive wavenumber régimes which together describe the most relevant possibilities for vortex behavior. In the first of these, at relatively small wavenumbers, the mode is inviscid in character; as the vortex wavenumber increases so the spatial amplification rate of the vortices increases until viscous effects become significant and the growth rate begins to diminish. As the wavenumber increases yet further so the vortex is completely stabilized. Here we discuss the corresponding structures which may exist within a separated flow and the most significant result we find is that the maximum growth rate of a mode in this type of flow is actually greater than that which occurs when the flow has not separated. In addition, the inviscid modes are shown to have a far more complicated configuration than within an attached boundary layer and, indeed, their structure can only be completely determined by implementation of numerical procedures.     

On the necessary conditions for optimal control of retarded systems

In this paper we derive necessary conditions, in the form of a maximum principle, for the optimal control of nonlinear, finitely retarded functional differential equations with function-space boundary conditions. We establish these conditions in a setting which guarantees the existence of regular multipliers, admits pointwise control constraints, and, with added restrictions, ensures nontriviality of the multipliers.The majority of this work was done while the first author was guest at the Institute für Numerische und Angewandte Mathematik der Universität Göttingen, Göttingen, BRD.     

On the rate of superlinear convergence of a class of variable metric methods

Summary This paper considers a class of variable metric methods for unconstrained minimization. Without requiring exact line searches each algorithm in this class converges globally and superlinearly on convex functions. Various results on the rate of the superlinear convergence are obtained.Dedicated to Professor Dr. H. Görtler on the occasion of his seventieth birthday     

Stability of the flow around a cylinder: The spin-up problem

In this paper the evolution of vortical disturbances withina boundary layer occurring at a circular cylinder is discussed.As the body starts to spin around its generator a temporallygrowing layer results and the underlying centrifugal natureof this system allows Taylor-Görtler vortices to develop.The initiation of these modes is caused by axially symmetricwall imperfections, and so this constitutes a receptivity problem.For vortices of order one wavenumber a vortex wedge is formedand the final structure of this mode is determined by a right-handbranch calculation. The inviscid limit of the calculation isalso discussed and this may well be relevant to modes with orderone wavenumber introduced after the layer has partially evolved.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Counterexamples to Hölmgren's uniqueness for analytic non linear Cauchy problems

Summary In contrast to Hölmgren uniqueness for linear equations and to the similar uniqueness result for first order scalar nonlinear equations, this paper gives elementary examples of analytic nonlinear higher order equations, for which uniqueness ofC solutions to the non characteristic Cauchy problem fails.Oblatum 11-V-1992 & 19-X-1992     

A Note on High/Low-Wave-Number Interactions in Spatially Discrete Parabolic Equations

Present address: School of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. We describe an instability introduced by the spatial discretizationof reaction-diffusion equations. The mechanism is a nonlinearinteraction between high and low wave-number modes in the discreteequations. In partial differential equations which exhibit strongtemporal growth, a parasitic high-wave-number mode is stimulated,through aliasing, by a physically meaningful low-wave-numbermode. We analyse the interaction using phase-plane techniquesand present complementary numerical results.