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A systematic theory is developed to study the nonlinear spatial evolution of the resonant triad in Blasius boundary layers. This triad consists of a plane wave at the fundamental frequency and a pair of symmetrical, oblique waves at the subharmonic frequency. A low-frequency asymptotic scaling leads to a distinct critical layer wherein nonlinearity first becomes important, and the critical layer's nonlinear, viscous dynamics determine the development of the triad.The plane wave initially causes double-exponential growth of the oblique waves. The plane wave, however, continues to follow the linear theory, even when the oblique waves’ amplitude attains the same order of magnitude as that of the plane wave. However, when the amplitude of the oblique waves exceeds that of the plane wave by a certain level, a nonlinear stage comes into effect in which the self-interaction of the oblique waves becomes important. The self-interaction causes rapid growth of the phase of the oblique waves, which causes a change of the sign of the parametric-resonance term in the oblique-waves amplitude equation. Ultimately this effect causes the growth rate of the oblique waves to oscillate around their linear growth rate. Since the latter is usually small in the nonlinear regime, the net outcome is that the self-interaction of oblique waves causes the parametric resonance stage to be followed by an ‘oscillatory’ saturation stage.

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The spatial interactions between a fundamental instability wave and its subharmonics in a turbulent round jet are studied for ‘natural’ or forced exit conditions. Time-averaging and conditional-averaging techniques are used to split each flow component into a mean one, a random turbulence one and several wave-like coherent-structure components at fundamental and subharmonic frequencies. The energy equations for the flow components are derived and integrated across the jet. Shape assumptions regarding the radial distributions of each flow component are used to obtain a set of nonlinear ordinary differential equations representing the energy interactions between the coherent components, while interacting with the mean flow and with the background turbulence. Vortex pairing is viewed here as occurring when the subharmonic absorbs energy from the fundamental and from the mean flow and exceeds the fundamental's level to become the dominant instability component. At the proper initial phase difference between the subharmonic and fundamental only the first subharmonic was found to amplify if the fundamental Strouhal number based on diameter is in the range of 0.6–1.0. For higher Strouhal numbers, several subharmonics can amplify. The pairing location moves closer to the nozzle exit with increasing excitation Strouhal number. The time-averaged coherent Reynolds stresses exhibit regions of sign change, indicating a reversal in the direction of energy transfer between the mean flow and the coherent components.

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