Two time-periodic solutions with genuine three-dimensional structure are numerically discovered for the incompressible Navier–Stokes equation of a constrained plane Couette flow. One solution with strong variation in spatial and temporal structure exhibits a full regeneration cycle, which consists of the formation and breakdown of streamwise vortices and low-velocity streaks; the other one, of gentle variation, represents a spanwise standing-wave motion of low-velocity streaks. These two solutions are unstable and the corresponding periodic orbits in the phase space are connected with each other. A turbulent state wanders around the strong one for most of the time except for occasional escapes from it. As a result, the mean velocity profile and the root-mean-squares of velocity fluctuations of the plane Couette turbulence agree very well with the temporal averages of those of this periodic motion. After an occasional escape from the strong solution, the turbulent state reaches the gentle periodic solution and returns. On the way back, it experiences an overshoot accompanied by strong turbulence activity like an intermittent bursting phenomenon.

Recent remarkable progress in computing power and numerical analysis is enabling us to fill a gap in the dynamical systems approach to turbulence. One of the significant advances in this respect has been the numerical discovery of simple invariant sets, such as nonlinear equilibria and periodic solutions, in well-resolved Navier--Stokes flows. This review describes some fundamental and practical aspects of dynamical systems theory for the investigation of turbulence, focusing on recently found invariant solutions and their significance for the dynamical and statistical characterization of low-Reynolds-number turbulent flows. It is shown that the near-wall regeneration cycle of coherent structures can be reproduced by such solutions. The typical similarity laws of turbulence, i.e. the Prandtl wall law and the Kolmogorov law for the viscous range, as well as the pattern and intensity of turbulence-driven secondary flow in a square duct can also be represented by these simple invariant solutions.Comment: To appear in Annual Review of Fluid Mechanics, Vol. 44, 201

The behaviour of turbulent shear flow over a mass-neutral permeable wall is studied numerically. The transpiration is assumed to be proportional to the local pressure fluctuations. It is first shown that the friction coefficient increases by up to 40% over passively porous walls, even for relatively small porosities. This is associated with the presence of large spanwise rollers, originating from a linear instability which is related both to the Kelvin–Helmholtz instability of shear layers, and to the neutral inviscid shear waves of the mean turbulent profile. It is shown that the rollers can be forced by patterned active transpiration through the wall, also leading to a large increase in friction when the phase velocity of the forcing resonates with the linear eigenfunctions mentioned above. Phase-lock averaging of the forced solutions is used to further clarify the flow mechanism. This study is motivated by the control of separation in boundary layers.

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Near-wall turbulence in the buffer region of Couette and Poiseuille flows is characterized in terms of recently-found nonlinear three-dimensional solutions to the incompressible Navier-Stokes equations for wall-bounded shear flows. The data suggest that those solutions can be classified into two families, of which one is dominated by streamwise vortices, and the other one by streaks. They can be associated with the upper and lower branches of the equilibrium solutions for Couette flow found by Nagata ["Three-dimensional finite-amplitude solutions in plane Couette flow: Bifurcation from infinity," J. Fluid Mech. 217, 519 (1990)]. The quiescent structures of near-wall turbulence are shown to correspond to the vortex-dominated family, but evidence is presented that they burst intermittently both in minimal and in fully turbulent flows. The intensity and period of the bursts are Reynolds-number dependent, but they saturate at high enough Reynolds numbers. The time-periodic exact solution found for Couette flow by Kawahara and Kida ["Periodic motion embedded in plane Couette turbulence: Regeneration cycle and burst," J. Fluid Mech. 449, 291 (2001)] can be used as a simplified model for the bursting process.

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 measures around 190 wall units, roughly independently of the Reynolds number. Furthermore, we present a characterization of the flow state at marginal Reynolds numbers which substantially differs from the fully turbulent one. In particular, the marginal state exhibits a 4-vortex secondary flow structure alternating in time whereas the fully turbulent one presents the usual 8-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 linear inviscid instability of an infinitely thin vortex sheet, periodically corrugated with finite amplitude along the spanwise direction, is investigated analytically. Two types of corrugations are studied, one of which includes the presence of an impermeable wall. Exact eigensolutions are found in the limits of very long and of very short wavelengths. The intermediate-wavenumber range is explored by means of a second-order asymptotic series and by limited numerical integration. The sheets are unstable to both sinuous and varicose disturbances. The former are generally found to be more unstable, although the difference only appears for finite wavelengths. The effect of the corrugation is shown to be stabilizing, although in the wall-bounded sheet the effect is partly compensated by the increase in the distance from the wall. The controlling parameter in that case appears to be the minimum separation from the sheet valley to the wall. The instability is traced to a pair of oblique Kelvin-Helmholtz waves in the flat-sheet limit, but the eigenfunctions change character both as the corrugation is made sharper and as the wall is approached, becoming localized near the crests and valleys of the corrugation. The study is motivated by the desire to understand the behaviour of lifted low-speed streaks in wall-bounded flows, and it is shown that the spatial structure of the fundamental sinuous eigenmode is remarkably similar to previously known three-dimensional nonlinear equilibrium solutions in both plane Couette and Poiseuille flows.