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The nonlinear behaviour of a constant vorticity layer at a wall

The so-called ‘water-bag’ method is used to study the behaviour of a two-dimensional inviscid layer of constant vorticity ω and of mean thickness δ adjacent to a wall with slip at the wall. A nonlinear initial-value equation is derived which describes the motion of the material interface separating the rotational fluid within the layer from the irrotational free stream, for the case where this interface is subject to streamwise cyclic disturbances to its undisturbed shape. A linearized solution to this equation shows that a sinusoidal disturbance of wavelength λ propagates as one mode of a neutrally stable Kelvin-Helmholtz wave with velocity ωλ[1 − exp (−4πδ/λ)]/4π relative to the fluid at infinity. Numerical solutions of the full nonlinear equation for a range of wavelengths and finite disturbance amplitudes indicate different behaviour. For sufficiently large amplitude the interface valleys evolve into long re-entrant wedges of irrotational fluid which are ‘entrained’ into the layer and which are separated from the free stream by lobes or bulges of rotational fluid. This single-mode nonlinear interfacial distortion could be generated over a broad wavelength range with no indication of preferential scaling based on δ. It is suggested that the interface behaviour bears distinct resemblance to flow features observed at the interface between turbulent and non-turbulent fluid in recent smoke-in-air flow-visualization studies of the outer part of a constant pressure turbulent boundary layer. The calculated rotational fluid lobe velocities, which are not very different from the equivalent linearized wave velocities, are found to be in reasonable agreement with the few existing measurements of the velocity of bulges at the turbulent–nonturbulent fluid interface, while the computed velocity field in the lobe is in qualitative agreement with the general flow pattern observed in experiments. In the absence of a preferred scale or range of scales for the development of the interfacial distortion, however, it is concluded that the present results cannot be interpreted as supporting the hypothesis of the presence of largescale coherent motions in the outer part of the layer.