We have performed direct numerical simulations of turbulent flows in a square duct considering a range of Reynolds numbers spanning from a marginal state up to fully developed turbulent states at low Reynolds numbers. The main motivation stems from the relatively poor knowledge about the basic physical mechanisms that are responsible for one of the most outstanding features of this class of turbulent flows: Prandtl's secondary motion of the second kind. In particular, the focus is upon the role of flow structures in its generation and characterization when increasing the Reynolds number. We present a two-fold scenario. On the one hand, buffer layer structures determine the distribution of mean streamwise vorticity. On the other hand, the shape and the quantitative character of the mean secondary flow, defined through the mean cross-stream function, are influenced by motions taking place at larger scales. It is shown that high velocity streaks are preferentially located in the corner region (e.g. less than 50 wall units apart from a sidewall), flanked by low velocity ones. These locations are determined by the positioning of quasi-streamwise vortices with a preferential sign of rotation in agreement with the above described velocity streaks' positions. This preferential arrangement of the classical buffer layer structures determines the pattern of the mean streamwise vorticity that approaches the corners with increasing Reynolds number. On the other hand, the centre of the mean secondary flow, defined as the position of the extrema of the mean cross-stream function (computed using the mean streamwise vorticity), remains at a constant location departing from the mean streamwise vorticity field for larger Reynolds numbers, i.e. it scales in outer units. This paper also presents a detailed validation of the numerical technique including a comparison of the numerical results with data obtained from a companion experiment.

The statistical properties are presented for the direct numerical simulation of a self-similar adverse pressure gradient (APG) turbulent boundary layer (TBL) at the verge of separation. The APG TBL has a momentum thickness-based Reynolds number range from $Re_{\unicode[STIX]{x1D6FF}_{2}}=570$ to 13 800, with a self-similar region from $Re_{\unicode[STIX]{x1D6FF}_{2}}=10\,000$ to 12 300. Within this domain the average non-dimensional pressure gradient parameter $\unicode[STIX]{x1D6FD}=39$ , where for a unit density $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FF}_{1}P_{\!e}^{\prime }/\unicode[STIX]{x1D70F}_{w}$ , with $\unicode[STIX]{x1D6FF}_{1}$ the displacement thickness, $\unicode[STIX]{x1D70F}_{w}$ the mean shear stress at the wall and $P_{\!e}^{\prime }$ the far-field pressure gradient. This flow is compared with previous zero pressure gradient and mild APG TBL ( $\unicode[STIX]{x1D6FD}=1$ ) results of similar Reynolds number. All flows are generated via the direct numerical simulation of a TBL on a flat surface with far-field boundary conditions tailored to apply the desired pressure gradient. The conditions for self-similarity, and the appropriate length and velocity scales, are derived. The mean and Reynolds stress profiles are shown to collapse when non-dimensionalised on the basis of these length and velocity scales. As the pressure gradient increases, the extent of the wake region in the mean streamwise velocity profiles increases, whilst the extent of the log-layer and viscous sublayer decreases. The Reynolds stress, production and dissipation profiles of the APG TBL cases exhibit a second outer peak, which becomes more pronounced and more spatially localised with increasing pressure gradient. This outer peak is located at the point of inflection of the mean velocity profiles, and is suggestive of the presence of a shear flow instability. The maximum streamwise velocity variance is located at a wall normal position of $\unicode[STIX]{x1D6FF}_{1}$ of spanwise wavelength of $2\unicode[STIX]{x1D6FF}_{1}$ . In summary as the pressure gradient increases the flow has properties less like a zero pressure gradient TBL and more akin to a free shear layer.

A direct numerical simulation of turbulent flow in a straight square duct was performed in order to determine the minimal requirements for self-sustaining turbulence. It was found that turbulence can be maintained for values of the bulk Reynolds number above approximately 1100, corresponding to a friction-velocity-based Reynolds number of 80. The minimum value for the streamwise period of the computational domain is around 190 wall units, roughly independently of the Reynolds number. We present a characterization of the flow state at marginal Reynolds numbers which substantially differs from the fully turbulent one: the marginal state exhibits a four-vortex secondary flow structure alternating in time whereas the fully turbulent one presents the usual eight-vortex pattern. It is shown that in the regime of marginal Reynolds numbers buffer-layer coherent structures play a crucial role in the appearance of secondary flow of Prandtl's second kind.

The three-dimensional vortex clusters, and the structures based on the quadrant classification of the intense tangential Reynolds stress (Qs), are studied in direct numerical simulations of statistically stationary homogeneous shear turbulence (HST) at Taylor microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}\approx 50{-}250$ , with emphasis on comparisons with turbulent channels (CHs). The Qs and vortex clusters in HST are found to be versions of the corresponding detached (in the sense of del Álamo et al. (J. Fluid Mech., vol. 561 (2006), pp. 329–358)) structures in CHs, although statistically symmetrised with respect to the substitution of sweeps by ejections and vice versa. In turn, these are more symmetric versions of the corresponding attached Qs and clusters. In both flows, only co-gradient sweeps and ejections larger than the local Corrsin scale are found to couple with the shear. They are oriented anisotropically, and are responsible for carrying most of the total Reynolds stress. Most large eddies in CHs are attached to the wall, but it is shown that this is probably a geometric consequence of their size, rather than the reason for their dynamical significance. Most small Q structures associated with different quadrants are far from each other in comparison to their size, but those that are close to each other tend to form quasi-streamwise trains of groups of a sweep and an ejection paired side by side in the spanwise direction, with a vortex cluster in between, generalising to three dimensions the corresponding arrangement of attached eddies in CHs. These pairs are organised around an inclined large-scale conditional vortex ‘roller’, and it is shown that the composite structure tends to be located at the interface between high- and low-velocity streaks, as well as in strong ‘co-gradient’ shear layers that separate streaks of either sign in which velocity is more uniform. It is further found that the conditional rollers are terminated by ‘hooks’ reminiscent of hairpins, both upright and inverted. The inverted hook weakens as the structures approach the wall, while the upright one changes little. At the same time, the inclination of the roller with respect to the mean velocity decreases from $45^{\circ }$ in HST to quasi-streamwise for wall-attached eddies. Many of these observations are generalised to intense Reynolds stresses formed with different pairs of velocity components, and it is shown that most properties of the small structures can be traced to their definitions, rather than to their dynamics. It is concluded that the larger Reynolds-stress structures are associated with shear turbulence, rather than with the presence of a wall, while the smaller ones are generic to turbulence in general, whether sheared or not.

Statistically stationary and homogeneous shear turbulence (SS-HST) is investigated by means of a new direct numerical simulation code, spectral in the two horizontal directions and compact-finite-differences in the direction of the shear. No remeshing is used to impose the shear-periodic boundary condition. The influence of the geometry of the computational box is explored. Since HST has no characteristic outer length scale and tends to fill the computational domain, long-term simulations of HST are “minimal” in the sense of containing on average only a few large-scale structures. It is found that the main limit is the spanwise box width, L_z, which sets the length and velocity scales of the turbulence, and that the two other box dimensions should be sufficiently large (L_x ≳ 2L_z, L_y ≳ L_z) to prevent other directions to be constrained as well. It is also found that very long boxes, L_x ≳ 2L_y, couple with the passing period of the shear-periodic boundary condition, and develop strong unphysical linearized bursts. Within those limits, the flow shows interesting similarities and differences with other shear flows, and in particular with the logarithmic layer of wall-bounded turbulence. They are explored in some detail. They include a self-sustaining process for large-scale streaks and quasi-periodic bursting. The bursting time scale is approximately universal, ∼20S⁻¹, and the availability of two different bursting systems allows the growth of the bursts to be related with some confidence to the shearing of initially isotropic turbulence. It is concluded that SS-HST, conducted within the proper computational parameters, is a very promising system to study shear turbulence in general.