Resonant growth of three-dimensional disturbances in plane Poiseuille flow

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Abstract
A linear mechanism for growth of three-dimensional perturbations on plane Poiseuille flow is investigated. The mechanism, resonant forcing of vertical vorticity waves by Tollmien–Schlichting waves, leads to an algebraic growth for small times. Eventually, viscous damping becomes dominant and the disturbance decays. The resonance occurs only at discrete points in the wave-number space. Nine resonances have been investigated. For these, the phase velocities range from 0[sdot ]67 to 0[sdot ]81 of the centre-line velocity. The lowest Reynolds number for which the resonance can occur is 25. The strongest resonance appears only above a Reynolds number of 341. Also, two cases of degeneracy in the Orr–Sommerfeld dispersion relationship have been found.