Asymptotic description of transitional and turbulent flows: effects of surface roughness on the boundary layer and the evolution of coherent structures in free shear flows

Abstract

In this thesis we investigate, using high-Reynolds-number asymptotic techniques, three fluid dynamics problems linked to transition and turbulence. The first problem concerns the impact of spanwise periodic, streamwise elongated surface roughness elements on the boundary-layer stability. We are mainly interested in their effects on the so-called lower-branch modes, therefore, the spanwise spacing of the roughness is chosen to be comparable to the characteristic wavelength of the instability modes which is on the triple-deck scale. The streamwise length is much longer, consistent with experimental setups. By appropriate rescaling, a nonlinear set of boundary-layer equations are derived from the generic triple-deck theory. As the pressure is completely determined by the roughness shape to leading order, the governing equations can be solved by an efficient marching method to compute the streaky flow. The instability of the streaky flow is reduced to a one-dimensional eigenvalue problem in the spanwise direction. The instability is found to be controlled by the (spanwise periodic) streamwise wall shear. The numerical results are then compared to existing experimental data for which a good qualitative agreement is obtained. The second problem investigates the nonlinear dynamics of coherent structures in the turbulent mixing layer. Experimental results have shown that these large-scale motions share many characteristics with instability waves. Most of the characteristics can be predicted by a stability analysis of the mean flow. The instantaneous flow field is decomposed into the mean flow, coherent and incoherent fluctuations. The problem is closed by applying appropriate turbulence closures to the so-called modulated Reynolds stress transport equations. The nonlinear critical-layer theory for laminar-flow instabilities is adapted to develop a mathematical theory describing the evolution of coherent structures. With the present high-level turbulence closure model, the effect of fine-grained turbulence on the coherent structures is shown to appear as novel dispersion as well as anisotropic diffusion, in contrast to isotropic diffusion in the gradient-type models. The theoretical predictions capture the main nonlinear features of the coherent structures. Lastly, we apply an existing weakly-nonlinear theory to the evolution of coherent structures in the turbulent wake, but unlike previous works, we use a newly observed non-equilibrium turbulent dissipation scaling law that holds in the region where energetic coherent structures are present, and is different from that linked to the classical Richardson–Kolmogorov equilibrium cascade.