A model is proposed for the one-dimensional spectrum and streamwise Reynolds stress in pipe flow for arbitrarily large Reynolds numbers. Constructed in wavenumber space, the model comprises four principal contributions to the spectrum: streaks, large-scale motions, very-large-scale motions and incoherent turbulence. It accounts for the broad and overlapping spectral content of these contributions from different eddy types. The model reproduces well the broad structure of the premultiplied one-dimensional spectrum of the streamwise velocity, although the bimodal shape that has been observed at certain wall-normal locations, and the $-5/3$ slope of the inertial subrange, are not captured effectively because of the simplifications made within the model. Regardless, the Reynolds stress distribution is well reproduced, even within the near-wall region, including key features of wall-bounded flows such as the Reynolds number dependence of the inner peak, the formation of a logarithmic region, and the formation of an outer peak. These findings suggest that many of these features arise from the overlap of energy content produced by both inner- and outer-scaled eddy structures combined with the viscous-scaled influence of the wall. The model is also used to compare with canonical turbulent boundary layer and channel flows, and despite some differences being apparent, we speculate that with only minor modifications to its coefficients, the model can be adapted to these flows as well.

The wall dependence of length scales used to describe large- and small-scale structures of turbulence is examined using highly resolved experiments in zero-pressure-gradient turbulent boundary layers and pipe flows spanning the range $2000< Re_\tau <37\ 700$. Of particular interest is the influence of external intermittency on the scaling of these length scales. It is found that when suitable scaling parameters are selected and external intermittency is accounted for, the dissipative motions follow inner scaling even into the outer-scaled regions of the flow, and that certain large-scale descriptions follow outer scaling even in the inner-scaled regions of the flow. The wall dependence is the same for both internal pipe and external boundary layer flows, and the different length scales can be related to recognizable features in the longitudinal wavenumber spectrum.

Direct numerical simulations are performed for incompressible, turbulent channel flow over a smooth wall and different sinusoidal wall roughness configurations at a constant $Re_\tau = 720$. Sinusoidal walls are used to study the effects of well-defined geometric features of roughness-amplitude, $a$, and wavelength, $\lambda$, on the flow. The flow in the near-wall region is strongly influenced by both $a$ and $\lambda$. Establishing appropriate scaling laws will aid in understanding the effects of roughness and identifying the relevant physical mechanisms. Using inner variables and the roughness function to scale the flow quantities provides support for Townsend's hypothesis, but inner scaling is unable to capture the flow physics in the near-wall region. We provide modified scaling relations considering the dynamics of the shear layer and its interaction with the roughness. Although not a particularly surprising observation, this study provides clear evidence of the dependence of flow features on both $a$ and $\lambda$. With these relations, we are able to collapse and/or align peaks for some flow quantities and, thus, capture the effects of surface roughness on turbulent flows even in the near-wall region. The shear-layer scaling supports the hypothesis that the physical mechanisms responsible for turbulent kinetic energy production in turbulent flows over rough walls are greatly influenced by the shear layer and its interaction with the roughness elements. Finally, a semiempirical model is developed to predict the contribution of pressure and skin friction drag on the roughness element based purely on its geometric parameters and the corresponding shear-layer velocity scale.