Direct numerical simulations are used to examine large-scale motions with a streamwise length$2\sim 4h$($h$denotes the channel half-width) in the logarithmic and outer regions of a turbulent channel flow. We test a minimal ‘streamwise’ flow unit (Toh & Itano,J. Fluid Mech., vol. 524, 2005, pp. 249–262) (or MSU) for larger Kármán numbers ($h^{+}=395$and 1020) than in the original work. This flow unit consists of a sufficiently long (${L_{x}}^{+}\approx 400$) streamwise domain to maintain near-wall turbulence (Jiménez & Moin,J. Fluid Mech., vol. 225, 1991, pp. 213–240) and a spanwise domain which is large enough to represent the spanwise behaviour of inner and outer structures correctly; as$h^{+}$increases, the streamwise extent of the MSU domain decreases with respect to$h$. Particular attention is given to whether the spanwise organization of the large-scale structures may be represented properly in this simplified system at sufficiently large$h^{+}$and how these structures are associated with the mean streamwise velocity$\overline{U}$. It is shown that, in the MSU, the large-scale structures become approximately two-dimensional at$h^{+}=1020$. In this case, the streamwise velocity fluctuation$u$is energized, whereas the spanwise velocity fluctuation$w$is weakened significantly. Indeed, there is a reduced energy redistribution arising from the impaired global nature of the pressure, which is linked to the reduced linear–nonlinear interaction in the Poisson equation (i.e. the rapid pressure). The logarithmic dependence of$\overline{ww}$is also more evident due to the reduced large-scale spanwise meandering. On the other hand, the spanwise organization of the large-scale$u$structures is essentially identical for the MSU and large streamwise domain (LSD). One discernible difference, relative to the LSD, is that the large-scale structures in the MSU are more energized in the outer region due to a reduced turbulent diffusion. In this region, there is a tight coupling between neighbouring structures, which yields antisymmetric pairs (with respect to centreline) of large-scale structures with a spanwise spacing of approximately$3h$; this is intrinsically identical with the outer energetic mode in the optimal transient growth of perturbations (del Álamo & Jiménez,J. Fluid Mech., vol. 561, 2006, pp. 329–358).
Nonlinear optimal large-scale structures in turbulent channel flow
