The spectral-element method (SEM) is used to study wall-bounded turbulent flowsin moderately complex geometries. The first part of the thesis is devoted to simulations of canonical flow cases, such as temporal K-type transitionand turbulent channel flow, to investigate general resolution requirements and computational efficiency of the numerical code nek5000. Large-eddy simulation (LES) is further performed of a plane asymmetric diffuser flow with an opening angle of 8.5 degrees, featuring turbulent flow separation. Good agreement with numerical studies of Herbst (2007) is obtained, and it is concluded that the use of a high-order method is advantageous for flows featuring pressure-induced separation. Moreover, it is shown, both a priori on simpler model problems and a posteriori using the full Navier--Stokes equations, that the numerical instability associated with SEM at high Reynolds numbers is cured either by employing over-integration (dealiasing) or a filter-based stabilisation, thus rendering simulations of moderate to high Reynolds number flows possible. The second part of the thesis is devoted to the first direct numerical simulation (DNS) of a truly three-dimensional, turbulent and separated diffuser flow at Re = 10 000 (based on bulk velocity and inflow-duct height), experimentally investigated by Cherry et al. (2008). The massively parallel capabilities of the spectral-element method are exploited by running the simulations on up to 32 768 processors. Very good agreement with experimental mean flow data is obtained and it is thus shown that well-resolved simulations of complex turbulent flows with high accuracy are possible at realistic Reynolds numberseven in complicated geometries. An explanation for the discovered asymmetry of the mean separated flow is provided and itis demonstrated that a large-scale quasi-periodic motion is present in the diffuser. In addition, a new diagnostic measure, based on the maximum vorticity stretching component in every spatial point, is designed and tested in a number of turbulent and transitional flows. Finally, Koopman mode decomposition is performed of a minimal channel flow and compared to classical proper orthogonal decomposition (POD). / QC 20111206