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.     

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.     

Instabilities of a flexible surface in supersonic flow

The stability of a supersonic boundary layer above a flexiblesurface is considered in the limit of large Reynolds numberand for Mach numbers O(1). Asymptotic theory of viscous–inviscidinteraction has been used for this purpose. We found that fora simple elastic surface, for which deflections are proportionalto local pressure differences, the boundary-layer flow remainsstable as it is for a rigid wall. However, when either dampingor surface inertia is included the flow becomes unstable. Moreover,in a certain range of wave numbers the boundary layer developsmore then one unstable mode. It is interesting that these modesare connected to one another via saddle points in the complex-frequencyplane. A more complex Kramer-type surface is also analysed andin some parameter ranges is found to permit the evolution ofunstable Tollmien–Schlichting waves. The neutral curvesare found for a variety of situations related to the parametersassociated with the flexible surface.     

On the Instability of Gortler Vortices to Nonlinear Travelling Waves

Author to whom correspondence should be addressed Recent theoretical work by Hall & Seddougui (1989) has shownthat strongly nonlinear high-wavenumber Görtler vorticesdeveloping within a boundary layer flow are susceptible to asecondary instability which takes the form of travelling wavesconfined to a thin region centred at the outer edge of the vortex.This work considered the case in which the secondary mode couldbe satisfactorily described by a linear stability theory, andin the current paper our objective is to extend this investigationof Hall & Seddougui (1989) into the nonlinear regime. Wefind that, at this stage, not only does the secondary mode becomenonlinear, but it also interacts with itself so as to modifythe governing equations for the primary Görtler vortex.In this case, then, the vortex and the travelling wave driveeach other, and indeed the whole flow structure is describedby an infinite set of coupled nonlinear differential equations.We undertake a Stuart-Watson type of weakly nonlinear analysisof these equations and conclude, in particular, that on thisbasis there exist stable flow configurations in which the travellingmode is of finite amplitude. Implications of our findings forpractical situations are discussed, and it is shown that thetheoretical conclusions drawn here are in good qualitative agreementwith available experimental observations.     

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.     

Free convection above a heated horizontal circular disk

The free convection boundary layer flow above a heated horizontal disk is considered. The equations of motion are solved numerically starting at the circumference of the disk where the flow is basically the same as that above a flat plate. The importance of the curvature effects increases as the centre is approached. It is shown that near the centre, the boundary-layer thickness is very large, and that the flow splits up into two distinct regions. There is a thin viscous region next to the disk of thickness ofO(r ^2/3), wherer measures distance from the centre and a thick outer inviscid region of thickness ofO(r ^?2/3).     

Compressible Modes of the Rotating-Disk Boundary-Layer Flow Leading to Absolute Instability

This work is devoted to the clarification of the viscous compressible modes particularly leading to absolute instability of the three-dimensional generalized Von Karman's boundary-layer flow due to a rotating disk. The infinitesimally small perturbations are superimposed onto the basic Von Karman's flow to achieve linearized viscous compressible stability equations. A numerical treatment of these equations is then undertaken to search for the modes causing absolute instability within the principle of Briggs–Bers pinching. Having verified the earlier incompressible and inviscid compressible results of [ 1–3 ], and also confirming the correct match of the viscous modes onto the inviscid ones in the large Reynolds number limit, the influences of the compressibility on the subject matter are investigated taking into consideration both the wall insulation and heat transfer. Results clearly demonstrate that compressibility, as the Mach number increases, acts in favor of stabilizing the boundary-layer flow, especially in the inviscid limit, as far as the absolute instability is concerned, although wall heating and insulation greatly enhances the viscous absolutely unstable modes (even more dramatic in the case of wall insulation) by lowering down the critical Reynolds number for the onset of instability, unlike the wall cooling.     

The linear stability of inviscid shear flow of a vibrationally excited diatomic gas

The problem of the linear stability of plane-parallel shear flows of a vibrationally excited compressible diatomic gas is investigated using a two-temperature gas dynamics model. The necessary and sufficient conditions for stability of the flows considered are obtained using the energy integrals of the corresponding linearized system for the perturbations. It is proved that thermal relaxation produces an additional dissipation factor, which enhances the flow stability. A region of eigenvalues of unstable perturbations is distinguished in the upper complex half-plane. Numerical calculations of the eigenvalues and eigenfunctions of the unstable inviscid modes are carried out. The dependence on the Mach number of the carrier stream, the vibrational relaxation time τ and the degree of non-equilibrium of the vibrational mode is analysed. The most unstable modes with maximum growth rate are obtained. It is shown that in the limit there is a continuous transition to well-known results for an ideal fluid as the Mach number and τ approach zero and for an ideal gas when τ → 0.     

Nonlinear high-wavenumber Bnard convection

Weakly nonlinear two-dimensional roll cells in Bnard convectionare examined in the limit as the wavenumber a of the roll cellsbecomes large. In this limit the second harmonic contributionsto the solution become negligible, and a flow develops wherethe fundamental vortex terms and the correction to the meanare determined simultaneously, rather than sequentially as inthe weakly nonlinear case. Extension of this structure to Rayleighnumbers O(a³) above the neutral curve is shown to be possible,with the resulting flow field having a form very similar tothat for strongly nonlinear vortices in a centripetally unstableflow. The flow in this strongly nonlinear regime consists ofa core region, and boundary layers of thickness O(a^–1)at the walls. The core region occupies most of the thicknessof the fluid layer and only mean terms and cos az terms playa role in determining the flow; in the boundary layer all harmonicsof the vortex motion are present. Numerical solutions of thewall layer equations are presented and it is also shown thatthe heat transfer across the layer is significantly greaterthan in the conduction state.     

Large-Amplitude Long-Wave Instability of a Supersonic Shear Layer

For sufficiently high Mach numbers, small disturbances on a supersonic vortex sheet are known to grow in amplitude because of slow nonlinear wave steepening. Under the same external conditions, linear theory predicts slow growth of long-wave disturbances to a thin supersonic shear layer. An asymptotic formulation that adds nonzero shear-layer thickness to the weakly nonlinear formulation for a vortex sheet is given here. Spatial evolution is considered for a spatially periodic disturbance having amplitude of the same order, in Reynolds number, as the shear-layer thickness. A quasi-equilibrium inviscid nonlinear critical layer is found, with effects of diffusion and slow growth appearing through a nonsecularity condition. Other limiting cases are also considered, in an attempt to determine a relationship between the vortex-sheet limit and the long-wave limit for a thin shear layer; there appear to be three special limits, corresponding to disturbances of different amplitudes at different locations along the shear layer.     

Flow Through Symmetrically Constricted Tubes

Fluid morion through a tube is discussed when there is a moderateor severe symmetric constriction at the wall, and the oncomingflow is fully developed. The Reynolds number R is assumed large.During a moderate constriction, where the typical slope isO(R^–), upstream separation can be provoked due to thedownstream pressure being transmitted back, through the inviscidcore flow. Separation can also occur after a point of maximumconstriction. Computations and analysis indicate that the upstreamseparation point is pushed increasingly ahead as the slope israised. The implication for a severe constriction, where isO(1), is that the flow separates at a large distance O(a lnR) upstream (a being the tubewidth) and produces a shear layerwhich, on nearing the constriction, reaches O(a) distances fromthe wall.     

Amplitude-dependent Stability of Boundary-layer Flow with a Strongly Non-linear Critical Layer

The amplitude-dependent neutral stability properties, mainlyof an accelerating boundary-layer flow, are studied theoreticallyfor large Reynolds numbers when the disturbance size is sufficientlylarge to provoke a strongly non-linear critical layer withinthe flow field. The theory has a rational basis aimed at a detailedunderstanding of the delicate physical balances controllingstability. It shows that when the fundamental disturbance size rises to O(R^-1/3, where R is the Reynolds number based on theboundary-layer thickness, the neutral wavelength shortens andthe wavespeed increases in such a way that they become comparablewith the typical thickness and speed, respectively, of the basicflow. In this Rayleigh-like situation a new (previously negligible)feature emerges, that of a substantial pressure variation acrossthe critical layer, which strongly affects the jump conditionson the Rayleigh solutions holding outside the critical layer.As a result of the strong non-linearity the total velocity jumpis affected non-linearly by the critical layer vorticity, whilein contrast the phase shift remains linearly dependent on thevorticity. Furthermore, it is shown that the phase shift, notthe total velocity jump, dictates the neutral stability criteria. Also, flow reversal occurs near the wall where the disturbanceis greater than the basic flow. The link between the viscouseffects in the wall layers and in the critical layer fixes theamplitude-dependence of the neutral modes throughout. As thedisturbance amplitude increases the critical layer with vorticitytrapped within it moves toward the edge of the boundary layerand is forced to leave the boundary layer when exceeds O(R^-1/3,if neutral stability is to be maintained. This departure israther abrupt, involving a dependence on (scaled amplitude)^–12.A study of the more practical application to temporally growingdisturbances should be interesting.     

The effect of buoyancy on upper-branch Tollmien-Schlichting waves

We investigate the effect of buoyancy on the upper-branch linearstability characteristics of an accelerating boundary-layerflow. The presence of a large thermal buoyancy force significantlyalters the stability structure. As the factor G (which is relatedto the Grashof number of the flow, and defined in Section 2)becomes large and positive, the flow structure becomes two layeredand disturbances are governed by the Taylor-Goldstein equation.The resulting inviscid modes are unstable for a large componentof the wavenumber spectrum, with the result that buoyancy isstrongly destabilizing. Restabilization is encountered at sufficientlylarge wavenumbers. For G large and negative the flow structureis again two layered Disturbances to the basic flow are nowgoverned by the steady Taylor—Goldstein equation in themajority of the boundary layer, coupled with a viscous walllayer. The resulting eigenvalue problem is identical to thatfound for the corresponding case of lower-branch Tollmien—Schlichtingwaves, thus suggesting that the neutral curve eventually becomesclosed in this limit.     

The Effect of Sidewalls on Overstable Convection in a Rotating Fluid

A simple two-dimensional model is used to demonstrate some interestingeffects which arise when Chandrasekhar's (1962) theory of overstableconvection in an infinite rotating fluid layer is modified totake account of lateral walls. The aim of the investigationis to determine how sidewalls aligned with the convective rollsaffect the critical Rayleigh number and frequency of oscillationand also how the overstable eigensolutions are related to thepreviously determined stationary solutions of the equations(Daniels, 1977). For containers of large aspect ratio, L, thecritical Rayleigh number for overstability is R_o+O(L^–1)(where R_o is the value for the infinite layer) and in the neighbourhoodof this single perturbed value it is found that there is aninfinite spectrum of overstable eigenvalues with frequencieswhich differ by O(L^–1). The O(L^–1) correction toR_o is determined analytically for the case of small Prandtlnumber and rapid rotation.     

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 Radiation of Short Surface Waves By a Finite Dock

A flat plate of finite width and infinite length lies on thesurface of a body of deep water and vibrates in such a way thatthe velocity distribution over the dock varies smoothly acrossits width and is simple harmonic in time. The amplitudes ofthe wave-trains radiated towards infinity are investigated inthe limit of waves having length small compared with the widthof the dock, by means of a formula expressing the wave amplitudein terms of a Green's function evaluated over the dock. Thisfunction is estimated in the short wave limit by an asymptoticapproximation which is uniformly valid over the dock exceptclose to the edges. Suitable edge corrections are presented,these being essentially the potentials in the correspondingproblems for docks of semi-infinite width. By this means itis found possible to derive the first two or three terms inan asymptotic development for the wave amplitudes, the firstterm agreeing with previous work, and the next term dependingon the nature of the velocity distribution V(x) near the edgesx = ±a. Explicit results are given for the cases whenV(x) is analytic, and when V(x) = O(a–x) near the edgex = a.     

Far downstream analysis for the Blasius boundary-layer stability problem

In this paper, we examine the large Reynolds number (Re) asymptoticstructure of the wave number in the Orr–Sommerfeld regionfor the Blasius boundary layer on a semi-infinite flat plategiven by Goldstein (1983, J. Fluid Mech., 127, 59–81).We show that the inclusion of the term which contains the leading-ordernon-parallel effects, at O(Re^{– 1/2}), leads to a non-uniformexpansion. By considering the far downstream form of each termin the asymptotic expansion, we derive a length scale at whichthe non-uniformity appears, and compare this position with theposition seen in plots of the wave number.     

零攻角小钝头钝锥高超音速绕流边界层的稳定性分析和转捩预报

研究了零攻角小钝头圆锥高超音速边界层的稳定性及转捩预测问题．小钝头的球头半径为0.5 mm,锥的半锥角为5°,来流马赫数为6．采用直接数值模拟方法得到了钝锥的基本流场,利用线性稳定性理论分析了等温壁面和绝热壁面条件下的第一、第二模态不稳定波,并用“e-N”方法对转捩位置进行了预测．在没有实验给出N值的情况下,暂取N为10．研究发现,壁面温度条件对于转捩位置有较大影响．绝热边界层的转捩位置比等温边界层的靠后．且尽管高马赫数下第二模态波的最大增长率远大于第一模态波的最大增长率,但绝热边界层的转捩位置是由第一模态不稳定波决定的．研究方法应能推广到有攻角的三维边界层流动的转捩预测．     

Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction

A spectral boundary-value problem is considered in a plane thick two-level junction Ω_ε formed as the union of a domain Ω₀ and a large number 2N of thin rods with thickness of order ε = O(N ⁻¹). The thin rods are split into two levels depending on their length. In addition, the thin rods from the indicated levels are ε-periodically alternating. The Fourier conditions are given on the lateral boundaries of the thin rods. The asymptotic behavior of the eigenvalues and eigenfunctions is investigated as ε → 0, i.e., when the number of thin rods infinitely increases and their thickness approaches zero. The Hausdorff convergence of the spectrum is proved as ε → 0, the leading terms of asymptotics are constructed, and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 195–216, February, 2006.     

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