For several applications there are advantages in writing turbulent flow equations in a coordinate frame aligned with the streamlines and several two-dimensional examples of this approach have appeared in the literature. In this paper, we extend this approach to general three-dimensional flows. We find that, in any flow that has a component of its vorticity aligned in the streamline direction, congruences of its streamlines do not form integrable manifolds. This limits the development of a streamline coordinate description of such flows, although some useful results can still be obtained. However, in the case of general three-dimensional complex-lamellar flows, where the mean velocity and mean vorticity are everywhere orthogonal, a complete streamline coordinate description can be derived. Furthermore, we show that general complex-lamellar flows are a good approximation to boundary layers and thin free shear layers. We derive the underlying true coordinate system for such flows, where the orthogonal coordinate surfaces are two stream surfaces and a modified potential surface. From this we obtain physical equations, where flow variables have the same dimensions they would have in a Cartesian coordinate frame. Finally, we show that rational approximations to these equations, which describe small-perturbation flows, contain some terms that have been ignored in previous applications and we detail some practical applications of the theory in modelling and analysis.

We compare the turbulence statistics of the canopy/roughness sublayer (RSL) and the inertial sublayer (ISL) above. In the RSL the turbulence is more coherent and more efficient at transporting momentum and scalars and in most ways resembles a turbulent mixing layer rather than a boundary layer. To understand these differences we analyse a large-eddy simulation of the flow above and within a vegetation canopy. The three-dimensional velocity and scalar structure of a characteristic eddy is educed by compositing, using local maxima of static pressure at the canopy top as a trigger. The characteristic eddy consists of an upstream head-down sweep-generating hairpin vortex superimposed on a downstream head-up ejection-generating hairpin. The conjunction of the sweep and ejection produces the pressure maximum between the hairpins, and this is also the location of a coherent scalar microfront. This eddy structure matches that observed in simulations of homogeneous-shear flows and channel flows by several workers and also fits with earlier field and wind-tunnel measurements in canopy flows. It is significantly different from the eddy structure educed over smooth walls by conditional sampling based only on ejections as a trigger. The characteristic eddy was also reconstructed by empirical orthogonal function (EOF) analysis, when only the dominant, sweep-generating head-down hairpin was recovered, prompting a re-evaluation of earlier results based on EOF analysis of wind-tunnel data. A phenomenological model is proposed to explain both the structure of the characteristic eddy and the key differences between turbulence in the canopy/RSL and the ISL above. This model suggests a new scaling length that can be used to collapse turbulence moments over vegetation canopies.