| Summary: | This thesis focuses on the use of systems theory to study the stability of viscous channel flow. In the first part of the thesis, the flow is modelled as a feedback connection between a linear dynamical subsystem and a memoryless nonlinear subsystem. After discretisation, the system is approximated by a finite-dimensional model and its global stability is analysed using the passivity theorems, which are extended from finite-dimensional Euclidean spaces to finite-dimensional Hilbert spaces. The passivity analysis leads to the Reynolds number below which the flow is globally stable and the flow can be unstable above this number because of possible energy amplification. The thesis then addresses inaccuracy of the numerical methods that are typically used in the literature to calculate the energy amplification in the flow. It is identified that the inaccuracy is caused by the presence of spurious eigenvalues with negative real parts of large magnitude and numerical integration errors. A remedy to this problem is given in the thesis. In the second part of the thesis, the effects of riblets on the linear stability and the energy amplification in channel flow are investigated numerically. Riblets are small protrusions aligned with the direction of the flow and it is known that the use of riblets on an aerodynamic body can result in significant reduction in drag, although the mechanism of drag reduction by riblets is not yet clear. Three types of riblets (sinusoidal, triangular and semi-circular ones) are considered and in order to use spectral methods to discretise the governing equations, the complex geometry associated with the riblets is transformed to a standard domain by a change of coordinates. The differential equations also have to be transformed into the new coordinate system and the presence of riblets poses a serious difficulty in representing the fluid flow by a state space model. To circumvent this difficulty, a novel formulation of the Navier-Stokes equations is derived.
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