Nonlinear modal interactions in a compressible boundary layer

• Main Author: Al-Salman, Adam
• Other Authors: Wu, Xuesong
• Published: Imperial College London 2003
• Subjects: 515.353
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.397152

It is well known that boundary layers in the supersonic regime can support multiple instability modes, including the well-documented first and second modes. The main aim of this thesis is to investigate nonlinear modal interactions. Since an instability mode attains its maximum magnitude in its critical layer (i.e. the region surrounding the position where the basic flow velocity is equal to the phase velocity), particularly effective interactions can take place among modes which share the same critical layer. As a first step, we solved the compressible Rayleigh equation, using parameters representing practical supersonic flows. Our calculations show that the resonant triad and phase-locked interactions can occur among purely first modes, or a certain combination of first and second modes. A new type of interaction between two or more modes which are frequency-locked, was identified. Each of these interactions has been studied in the so-called nonlinear nonequilibrium viscous critical-layer regime. For the resonant triad interaction, we derived a system of integro-partial-differential equations, which govern the spatial-temporal modulation of a triad, consisting of two oblique and one planar wavetrains. In the first stage, the amplitude of the planar wavetrain is governed by a linear equation. The oblique wavetrain amplitudes are governed by a set of coupled nonlinear equations, due to the quadratic interaction between the oblique and the planar wavetrains, which takes place in their common critical layers. These equations are solved for the two cases of interest where each wavetrain has a discrete spectrum or a narrow band continuous spectrum. Our results are able to capture experimental observations qualitatively. The disturbance enters the second stage once the planar wave has grown to such an extent that its self nonlinearity becomes significant. We find that the viscosity law can affect the form of the nonlinear term in the planar wave amplitude equation. As a result of this, the solution to this equation saturates. The previously super-exponential growth of the oblique wave is reduced to exponential growth. Similarly for the phase-locked interaction, we derived a system of integro-partial-differential equations, which govern the spatial-temporal modulation of a pair of wavetrains. Again in the first stage of the development, the planar mode is governed by a linear equation, but it interacts with the obhque mode to generate a large difference mode, which in turn interacts with the planar mode, to contribute a cubic nonlinear term to the amplitude equation of the oblique mode. These equations are solved for the case of interest where the wavetrains have a narrow band continuous spectra. The evolution of these waves in the second stage where the planar wave becomes nonlinear is also investigated. Next we studied the interaction between two oblique modes which are frequency-locked, in the sense that they all have the same streamwise wave number and frequency. This is a unique feature of supersonic flows, and may be relevant for the K-type of transition. We also investigate this type of interaction between two pairs of oblique modes. Finally we considered the streamwise-spanwise modulation of a nearly planar acoustic mode in a hypersonic boundary layer. We derived an evolution system consisting of an amplitude equation coupled to the two strongly nonlinear equations for the vorticity and temperature in the critical layer.