Branching of the Falkner-Skan solutions for λ&lt;0

Oskam, B.; Veldman, Arthur

doi:10.1007/bf00037732

Cited by 19 publications

(13 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The behaviour of (A 4) is essentially as given by (2.10) following a suitable substitution. Numerical results for the low-order exponential solution branches in n 6 0 are shown in figure 15; results of this form were first presented by Libby & Liu (1967), Oskam & Veldman (1982) and Brauner, Laine & Nicolaenko (1982).…”

Section: Discussionmentioning

confidence: 86%

Continua of states in boundary-layer flows

2002

View full text Add to dashboard Cite

We consider a class of three-dimensional boundary-layer flows, which may be viewed as an extension of the Falkner-Skan similarity form, to include a cross-flow velocity component, about a plane of symmetry. In general, this provides a range of threedimensional boundary-layer solutions, parameterized by a Falkner-Skan similarity parameter, n, together with a further parameter, Ψ ∞ , which is associated with a cross-flow velocity component in the external flow. In this work two particular cases are of special interest: for n = 0 the similarity equations possess a family of solutions related to the Blasius boundary layer; for n = 1 the similarity solution provides an exact reduction of the Navier-Stokes equations corresponding to the flow near a saddle point of attachment. It is known from the work of Davey (1961) that in this latter class of flow, a continuum of solutions can be found. The continuum arises (in general) because it is possible to find states with an algebraic, rather than exponential, behaviour in the far field. In this work we provide a detailed overview of the continuum states, and show that a discrete infinity of 'exponential modes' are smoothly embedded within the 'algebraic modes' of the continuum. At a critical value of the cross-flow, these exponential modes appear as a cascade of eigensolutions to the far-field equations, which arise in a manner analogous to the energy eigenstates found in quantum mechanical problems described by the Schrödinger equation.The presence of a discrete infinity of exponential modes is shown to be a generic property of the similarity equations derived for a general n. Furthermore, we show that there may also exist non-uniqueness of the continuum; that is, more than one continuum of states can exist, that are isolated for fixed n and Ψ ∞ , but which are connected through an unfolded transcritical bifurcation at a critical value of the cross-flow parameter, Ψ ∞ .The multiplicity of states raises the question of solution selection, which is addressed using two stability analyses that assume the same basic symmetry properties as the base flow. In one case we consider a steady, algebraic form in the 'streamwise' direction, whilst in the other a temporal form is assumed. In both cases it is possible to extend the analysis to consider a continuous spectrum of disturbances that decay algebraically in the wall-normal direction. We note some obvious parallels that exist between such stability analyses and the approach to the continua of states described earlier in the paper.We also discuss the appearance of analogous non-unique states to the Falkner-Skan equation in the presence of an adverse pressure gradient (i.e. n < 0) in an appendix. † Manchester Centre for Nonlinear Dynamics.‡ Current address:

show abstract

Section: Discussionmentioning

confidence: 86%

Continua of states in boundary-layer flows

2002

View full text Add to dashboard Cite

show abstract

“…The solutions obtained by Craven and Peletier exhibit a regular behaviour, strongly suggesting that for ~ > 1 a periodic solution exists as in the case )~ < -1 [2]. The present paper pursues this possibility.…”

Section: Introductionmentioning

confidence: 57%

“…f ' > 1 for some values of the argument ~/), emanate from a giant branching point at X = -1, f " ( 0 ) = , 1.08638; see Oskam and Veldman [2]. These solutions, which exhibit overshoot (i.e.…”

Section: Introductionmentioning

confidence: 97%

See 1 more Smart Citation

The role of periodic solutions in the Falkner-Skan problem for λ>0

1986

Self Cite

View full text Add to dashboard Cite

S u m m a r yThe Falkner-Skan equation f '" + if" + h(1 -f,2) = 0 is discussed for X > 0. Two types of solutions have been pursued: those satisfying f(0) = f'(0) = 0, f'(oe) = 1 and those being periodic. In both cases, numerical evidence is given for a rich structure of multiple solutions. Branching occurs for X = 1, 2, 3 ..... All solutions can be characterized by means of a special subset of periodic solutions.

show abstract

“…In particular with respect to the non-unique solutions [2], found reverse flow solutions for λ > 1. In addition [3], found a reverse flow solution for −0.1988 < λ < 0, [4] in particular considered reverse flow solutions for −0.5 < λ < −0.1988 and [5] investigated the solutions for λ < −1. More recently additional non-unique solutions of (1) have been found by considering flow past a stretched boundary by [6].…”

Section: Introductionmentioning

confidence: 99%

The stability of Falkner-Skan flows with several inflection points

Heeg¹,

Dijkstra²,

Zandbergen³

1999

Z. angew. Math. Phys.

View full text Add to dashboard Cite

Non-unique solutions to the Falkner-Skan equation with multiple inflection points have been investigated with respect to stability properties. A temporal stability analysis based on the Orr-Sommerfeld equation has been performed. Attention has been paid to the effect of the number of inflection points in these solutions on the stability properties. While the standard Falkner-Skan flow does not have any inflection points in a favourable pressure gradient, the first non-unique solution branch has two such points. The presence of these two inflection points implies a dramatic reduction of the critical Reynolds number by four orders of magnitude. Moreover, in base flows with more than two inflection points additional neutral waves occur with the same Reynolds number but with different wave numbers. Mathematics Subject Classification (1991).Stability of parallel flows, 76E05.

show abstract

Branching of the Falkner-Skan solutions for λ<0

Cited by 19 publications

References 19 publications

Continua of states in boundary-layer flows

Continua of states in boundary-layer flows

The role of periodic solutions in the Falkner-Skan problem for λ>0

The stability of Falkner-Skan flows with several inflection points

Contact Info

Product

Resources

About