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The Gröwth of Gortler vortices in boundary layers on concave walls is investigated. It is shown that for vortices of wavelength comparable to the boundary-layer thickness the appropriate linear stability equations cannot be reduced to ordinary differential equations. The partial differential equations governing the linear stability of the flow are solved numerically, and neutral stability is defined by the condition that a dimensionless energy function associated with the flow should have a maximum or minimum when plotted as a function of the downstream variable X. The position of neutral stability is found to depend on how and where the boundary layer is perturbed, so that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the Gortler problem, except for asymptotically small wavelengths. The results obtained are compared with previous parallel-flow theories and the small-wavelength asymptotic results of Hall (1982a, b), which are found to be reasonably accurate even for moderate values of the wavelength. The parallel-flow theories of the growth of Gortler vortices are found to be irrelevant except for the small-wavelength limit. The main deficiency of the parallel-flow theories is shown to arise from the inability of any ordinary differential approximation to the full partial differential stability equations to describe adequately the decay of the vortex at the edge of the boundary layer. This deficiency becomes intensified as the wavelength of the vortices increases and is the cause of the wide spread of the neutral curves predicted by parallel-flow theories. It is found that for a wall of constant radius of curvature a given vortex imposed on the flow can grow for at most a finite range of values of X. This result is entirely consistent with, and is explicable by the asymptotic results of, Hall (1982a).

The relationship between asymptotic descriptions of vortex–wave interactions and more recent work on ‘exact coherent structures’ is investigated. In recent years immense interest has been focused on so-called self-sustained processes in turbulent shear flows where the importance of waves interacting with streamwise vortex flows has been elucidated in a number of papers. In this paper, it is shown that the so-called ‘lower branch’ state which has been shown to play a crucial role in these self-sustained processes is a finite Reynolds number analogue of a Rayleigh vortex–wave interaction with scales appropriately modified from those for external flows to Couette flow, the flow of interest here. Remarkable agreement between the asymptotic theory and numerical solutions of the Navier–Stokes equations is found even down to relatively small Reynolds numbers, thereby suggesting the possible importance of vortex–wave interaction theory in turbulent shear flows. The relevance of the work to more general shear flows is also discussed.