Continua of states in boundary-layer flows

Abstract

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 three- dimensional 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.