Breakdown of boundary layers: (i) on moving surfaces; (ii) in semi-similar unsteady flow; (iii) in fully unsteady flow |
| |
| Authors: | J W Elliott F T Smith S J Cowley |
| |
| Institution: | 1. Mathematics Department , Imperial College , London , SW7 2BZ , U.K.;2. Department of Applied Mathematics and Theoretical Physics , Cambridge , CB3 9EW , U.K. |
| |
| Abstract: | Abstract The breakdown and separation or reattachment of boundary layers adjoining a mainstream are studied in the three related situations (i)-(iii) of the title. For (i) the classical steady boundary layer generally admits a logarithmic singularity in the displacement when breakdown occurs on a downstream-moving surface whereas the corresponding singularity for an upstream-moving surface can be logarithmic or of minus-one-sixth form. Conversely, the breakdown can be delayed to the onset of zero mainstream flow, in which case the displacement singularity is again logarithmic. In certain flows these singularities prove to be removable locally, yielding a breakaway separation or reattachment and including the first known successes of a classical strategy in describing large-scale separation. Other flows, by contrast, require an interactive strategy. Again, even on a fixed surface a breakdown different from Goldstein's can be produced if there is a moving section of surface further upstream. The application to (ii), semi-similar unsteady boundary layers, e.g. near an impulsively started wedge-like trailing edge, then follows readily and predicts analogous forms of singularity. The corresponding singularity in displacement predicted for fully unsteady classical boundary layers, (iii), occurs within a finite time and, like (i) (usually) and (ii), a three-tiered breakdown is involved at first. Subsequently interaction comes into play. Comparisons with numerical and/or earlier work are noted. In all three situations (i)-(iii), although the dynamics involved near breakdown, separation or reattachment are predominantly inviscid, the presence of small viscosity is of significance in enforcing smoothness of the local velocity profiles. |
| |
| Keywords: | |
|
|
