Title:

Nonlinear instability of Stokeslayers

The Stokeslayer generated by a sinusoidally oscillating flat plate in an infinite fluid is an important prototype of unsteady flows, and is known to be susceptible to highfrequency instability. According to linear theory (Tromans, 1977; Cowley, 1987), small disturbances can grow exponentially over part of a period and become neutral at some time. The present study considers nonlinear effects on the instability under a highReynoldsnumber assumption, i.e. R ^ 1. It is shown that in the vicinity of the neutral time, the small disturbances evolve from a linear stage to a nonlinear stage due to enhanced nonlinear interactions inside the critical layers  the neighbourhood of a level at which the phase velocity of the disturbance is equal to the basic flow velocity. Specifically, we have studied the evolution of a twodimensional wave, a pair of oblique waves and a resonant triad of waves. The temporalspatial modulation of nearplanar waves is also considered. For a twodimensional disturbance, it is shown that the evolution is controlled by nonlinear eff'ects once the linear growth rate has decreased from order one to 0(6?), where e is the magnitude of the disturbance. The amplitude equation turns out to be an integrodifferential one of Hickernell type (Hickernell 1984). We study the amplitude equation by seriesextension and numerical methods. For the inviscid case, i.e. e >> the solutions always develop a finitetime singularity. When viscosity is included, i.e. e ~ R~^, it is found that (a) viscosity generally delays the finitetime singularity, and (b) if viscosity is sufficiently large the singularity can be completely eliminated, with the result that the wave evolves to an equilibrium state, confirming the findings of Goldstein Sz Leib (1989). In particular, for the present problem it is found that for disturbances with wavenumbers in the range [0.6, 1.3], a finitetime singularity always occurs, no matter how large the scaled viscosity parameter is. If e < C i.e. the disturbance is relatively small, the amplitude equation can be reduced to a classical StuartLandau equation by taking a limit of the integrodifferential equation. The stability of the equilibria is also studied. For the case where the disturbance consists of a pair of oblique waves, the nonlinear evolution stage comes at a much earlier time, namely when linear growth rate is ), and it then evolves over a much faster time scale 0(e~3), as in Goldstein & Choi (1989). Mathematically this is due to a pole type of singularity in the outer solution for the streamwise and spanwise velocities, which is much stronger than the branchpoint singularity associated with Uyy{yc) ^ 0. We show that for the disturbances with orderone spanwise wavenumbers, the amplitude equation is the same as that of Goldstein & Choi(1989), although Uyy{yc) = 0 in their case. However, we point out that when threedimensionality is relatively weak, the fact that Uyy{yc) 7^ 0 in our problem does bring in a difference. Moreover, we obtain an amplitude equation for the case when viscosity effects are important in the leading order equations of the critical layers, thus extend the analysis of Goldstein & Choi (1989). At least under the inviscid hmit, the solution of the amplitude equation can develop a finitetime singularity, corresponding to the finitedistance singularity of Goldstein & Choi (1989). In addition, it is noted that the nonlinear interaction of waves can generate strong vortices of the same magnitude as the waves. We suggest that these vortices may be identified as those observed in experiments, and may have relevance to the streaky structure in a turbulent boundary layer. For the case when three waves form a resonanttriad, we show that when the amplitude of the planar and the obhque waves are C)(c3) and 0(e) respectively, a mutual interaction takes place and the disturbances are governed by a coupled integrodifferential equation system. These amplitude equations are significantly different from those of Raetz (1959), Craik (1971), Smith & Stewart (1987) in two important aspects. Firstly the local growth rate depends on the the whole history of the evolution, unlike a conventional resonant triad, where the local growth rate depends only on the instantaneous amplitudes of the disturbances (see Goldstein & Lee, 1990). Secondly the back reaction of the obhque waves on the 2D wave is accounted for by two cubic terms and one quartic term rather than by one quadratic term. The amplitudes of the twodimensional and the threedimensional waves can exhibit a finitetime singularity, the structure for which is proposed. The evolution equations obtained in our study are shown to be valid until the magnitude of the disturbance becomes order one; thus they provide a full description of the development of resonanttriad waves from their linear small amplitude stage towards the fullynonhnear orderoneamplitude stage. For the temporalspatial modulation case, i.e. the case when the amplitude of the disturbance is a slowlyvarying function of spanwise position as well as of time, it is shown that the development of the disturbance is controlled by criticallayer nonlinearity when its linear growth rate decreases to 0(es). Nonlinear interactions influence the evolution by producing a streamwise vortex and crossflow distortion. The modulation equation turns out to be an integropayimZdifferential one containing hi storydependent nonlinear terms. A novel feature of the amplitude equation is that the derivatives with respect to space, including the highest derivative, appear in the nonlinear terms. These terms are associated with threedimensionality, and hence represent vortex stretching and tilting effects. The possible properties of the amplitude equation are discussed. It is shown that a localized singularity may occur in a finite time. This provides a possible explanation for the focusing of streamwise vorticity and the formation of streaks. The finitetime singularity, which can occurs in all the four cases studied, suggests that explosive growth is induced by criticallayer nonlinear effects. Hence an initially small disturbance is further amplified by nonlinear effects; moreover this nonlinear growth can prevent the disturbance evolving into an equilibrium state as implied by linear theory. We suggest that this nonlinear blowup of highfrequency disturbances is related to the bursting phenomena observed in experiments, and can lead to transition to turbulence.
