Another route to the three-dimensional development of Tollmien-Schlichting waves with finite amplitude

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Abstract
The Tollmien-Schlichting waves appearing as a result of instability of laminar flows develop a three-dimensional configuration as the amplitude becomes large enough. A new explanation of this experimentally observed phenomenon is attempted on the basis of a resonance theory. It is shown that the existence of two-dimensional waves with finite amplitude can induce three-dimensional distortion with spanwise periodicity of the mean-flow field. Under a certain condition for resonance, the distortion grows, in proportion to the product of time and an exponential function of time, up to quite a large magnitude, and consequently interacts with the Tollmien-Schlichting waves to yield new three-dimensional travelling waves with the same streamwise wavenumber as the two-dimensional waves, and with the same spanwise wavenumber as the mean flow. The resulting flow field is of the peakvalley-splitting type, as observed often in experiments. The growth rate of the three-dimensional part in the mean flow depends significantly upon values of the spanwise wavenumber, suggesting that there is a preferred range of spanwise periodicity in the three-dimensional development of unstable laminar flows.

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