On a class of unsteady, non-parallel, three-dimensional disturbances to boundary-layer flows

Abstract

Steady, spatial, algebraically growing eigenfunctions are now known to occur in several important classes of boundary-layer flow, including two-dimensional hypersonic boundary layers and more recently in Blasius boundary layers subject to three-dimensional linearized disturbances, and in more general three-dimensional boundary layers. These spatial eigensolutions are particularly important and intriguing, given that they exist within the broad limits of the classical steady boundary-layer approximation, and as such are independent of Reynolds number.In this paper we make the natural extension to these previous (stability) analyses by incorporating the effects of unsteadiness into the model for treating disturbances to a quite general class of similarity-type boundary-layer flows. The flow disturbances are inherently non-parallel, but this effect is properly incorporated into the analysis.A further motivation for this paper is that Duck et al. (1999, 2000) have shown that by permitting a spanwise component of flow within a boundary layer of the appropriate form (in particular, growing linearly with the spanwise coordinate), it is found that new families of solutions exist – even the Blasius boundary layer has a three-dimensional ‘cousin’. Therefore a further aim of this paper is to assess the stability of the different solution branches, using the ideas introduced in this paper, to give some clues as to which of the solutions may be encountered experimentally.Several numerical methods are presented for tackling various aspects of the problem. It is shown that when algebraically growing, steady eigensolutions exist, their effect remains important in the unsteady context. We show how even infinitesimal, unsteady flow perturbations can provoke extremely large-amplitude flow responses, including in some cases truly unstable flow disturbances which grow algebraically downstream without bound in the linear context. There are some interesting parallels suggested therefore regarding mechanisms perhaps linked to bypass transition in an important class of boundary-layer flows.

Continua of states in boundary-layer flows

“…The first corresponds to a three-dimensional, steady class, which grows algebraically in the streamwise (x) direction, whilst retaining the same dependence as the base flow in the spanwise (z) direction; see (2.1). This type of disturbance has been investigated for the case of Falkner-Skan and other, related, three-dimensional boundary layers (albeit with no free-stream cross-flow velocity components) by Duck & Dry (2001) and Duck (2002). From a mathematical point of view, the attractive feature of this method (which is also related the work of Luchini (1996), is that the procedure is entirely rational within the context of the boundary-layer approximation.…”

“…We may therefore seek solutions in the form (6.8) assuming the amplitude parameter |ε| 1. To gain a simple heuristic estimate for the effect of temporal disturbances on the flow for other values of n (in particular n = 0) we also adopted this form for the general case; a similar procedure was used by Ridha (1992) (the proper procedure was described by Duck & Dry 2001).…”

On the growth (and suppression) of very short-scale disturbances in mixed forced–free convection boundary layers

“…those that do not exhibit breakdowns and those that do) are full and proper solutions of the complete equation set. Further, the work of Duck & Dry (2001), which was partly concerned with an initial value approach (both in the streamwise direction and temporally) does suggest that bounded solutions are fully realisable far downstream. Indeed, it is worth re-emphasizing that even the similarity form of the equations can exhibit non-uniqueness over a range of (negative) values of G 0 .…”

Separable Non-Parallel and Unsteady Flow Stability Problems

“…Griffond and Casalis 2001, Joslin 1996), or the perturbation flow field may be taken to have the same general form as the basic state(Duck and Dry 2001). Although the stability analysis restricted to perturbations of specific forms is not complete, it enables one to show that the flow is susceptible to a special kind of instability.…”

“…-(23) remain valid if we introduce the nondimensional variables, with the time scale 1/|b| and the correspondingly defined velocity scale. In the dimensionless equations (we will retain the same notation for the nondimensional variables), the parameter b takes one of the two values: b = 1 or b = −1, and ν is replaced by 1/Re where Re is the Reynolds number.Note that the solution (14) -(15) for b = 1 undergos the finite-time 'breakdown' (see, e.g.,Duck and Dry 2001, Hall et al 1992) and the 'normal mode' forms (21)-(22) are naturally adjusted to the description of the disturbed flow as the breakdown time t = 1 is approached.Next we will consider several two-point boundary value problems corresponding different specifications of the basic flow (14)-(19). (i) Axially symmetrical stagnation-point type flows.…”