Direct numerical simulations of a highly constrained plane Couette flow are employed to study the dynamics of the structures found in the near-wall region of turbulent flows. Starting from a fully developed turbulent flow, the dimensions of the computational domain are reduced to near the minimum values which will sustain turbulence. A remarkably well-defined, quasi-cyclic and spatially organized process of regeneration of near-wall structures is observed. This process is composed of three distinct phases: formation of streaks by streamwise vortices, breakdown of the streaks, and regeneration of the streamwise vortices. Each phase sets the stage for the next, and these processes are analysed in detail. The most novel results concern vortex regeneration, which is found to be a direct result of the breakdown of streaks that were originally formed by the vortices, and particular emphasis is placed on this process. The spanwise width of the computational domain corresponds closely to the typically observed spanwise spacing of near-wall streaks. When the width of the domain is further reduced, turbulence is no longer sustained. It is suggested that the observed spacing arises because the time scales of streak formation, breakdown and vortex regeneration become mismatched when the streak spacing is too small, and the regeneration cycle at that scale is broken.

A self-sustaining process conjectured to be generic for wall-bounded shear flows is investigated. The self-sustaining process consists of streamwise rolls that redistribute the mean shear to create streaks that wiggle to maintain the rolls. The process is analyzed and shown to be remarkably insensitive to whether there is no-slip or free-slip at the walls. A low-order model of the process is derived from the Navier–Stokes equations for a sinusoidal shear flow. The model has two unstable steady solutions above a critical Reynolds number, in addition to the stable laminar flow. For some parameter values, there is a second critical Reynolds number at which a homoclinic bifurcation gives rise to a stable periodic solution. This suggests a direct link between unstable steady solutions and almost periodic solutions that have been computed in plane Couette flow. It is argued that this self-sustaining process is responsible for the bifurcation of shear flows at low Reynolds numbers and perhaps also for controlling the near-wall region of turbulent shear flows at higher Reynolds numbers.

Nonlinear interactions in homogeneous turbulence are investigated using a decomposition of the velocity field in terms of helical modes. There are two helical modes per wave vector and thus eight fundamental triad interactions. These eight elementary interactions fit in only two classes, ‘‘R’’ (for ‘‘reverse’’) and ‘‘F’’ (for ‘‘forward’’), depending on whether the small-scale helical modes have helicities of the same or of the opposite sign. In a single-triad interaction, the large scale is unstable when the small-scale helical modes have helicities of opposite signs (class ‘‘F’’), and the medium scale is unstable otherwise (class ‘‘R’’). It is proposed that, on average, the triple correlations in a turbulent flow correspond to these unstable states. In the limit of nonlocal triads, where one leg is much smaller than the other two, the triadic interscale energy transfer is largest for interactions of class ‘‘R.’’ In that case, most of the energy flows locally in wave number, from the medium scale to the smallest, with a comparatively small feedback on the large scale. However, integrating over all scales in an inertial range, the net effect of nonlocal interactions of class ‘‘R’’ is a reverse energy cascade from small to large scales. All other interactions transfer energy upward in wave number. In local triads, this upward energy transfer occurs primarily between modes with helicities of the same sign, through catalytic interactions with a mode whose helicity has the opposite sign. The class ‘‘F’’ interactions transfer energy to the small scales and exist only in three dimensions. The physical processes associated to both classes of interactions are discussed. It is shown that the large local transfers due to nonlocal ‘‘R’’ interactions appear in pairs of opposite signs that nearly cancel each other and the net effect corresponds to an advection in wave space.

Forced turbulence in a rotating frame is studied using numerical simulations in a triply periodic box. The random forcing is three dimensional and localized about an intermediate wavenumber kf. The results show that energy is transferred to scales larger than the forcing scale when the rotation rate is large enough. The scaling of the energy spectrum approaches E(k)∝k−3 for k<kf. Almost all of the energy for k<kf lies in the two-dimensional (2D) plane perpendicular to the rotation z-axis, and thus the large-scale motions are quasi-2D with E(k)≈E(kh,kz=0), where kh and kz are, respectively, the horizontal and vertical components of the wavevector. The large scales consist of cyclonic vortices. Possible mechanisms responsible for the two-dimensionalization are discussed. The development of the 2D spectrum E(kh,kz=0)∝kh−3 is analogous to the dynamics of β-plane turbulence leading to the Rhines spectrum E(ky,kx=0)∝ky−5.

Three-dimensional steady states and traveling wave solutions of the Navier-Stokes equations are computed in plane Couette and Poiseuille flows with both free-slip and no-slip boundary conditions. They are calculated using Newton's method by continuation of solutions that bifurcate from a two-dimensional streaky flow then by smooth transformation ͑homotopy͒ from Couette to Poiseuille flow and from free-slip to no-slip boundary conditions. The structural and statistical connections between these solutions and turbulent flows are illustrated. Parametric studies are performed and the parameters leading to the lowest onset Reynolds numbers are determined. In all cases, the lowest onset Reynolds number corresponds to spanwise periods of about 100 wall units. In particular, the rigid-free plane Poiseuille flow traveling wave arises at Re ϭ44.2 for L x ϩ ϭ273.7 and L z ϩ ϭ105.5, in excellent agreement with observations of the streak spacing. A simple one-dimensional map is proposed to illustrate the possible nature of the ''hard'' transition to shear turbulence and connections with the unstable exact coherent structures.

Transition to turbulence in pipe flow is one of the most fundamental and longest-standing problems in fluid dynamics. Stability theory suggests that the flow remains laminar for all flow rates, but in practice pipe flow becomes turbulent even at moderate speeds. This transition drastically affects the transport efficiency of mass, momentum, and heat. On the basis of the recent discovery of unstable traveling waves in computational studies of the Navier-Stokes equations and ideas from dynamical systems theory, a model for the transition process has been suggested. We report experimental observation of these traveling waves in pipe flow, confirming the proposed transition scenario and suggesting that the dynamics associated with these unstable states may indeed capture the nature of fluid turbulence.

Lower branch coherent states in plane Couette flow have an asymptotic structure that consists of O(1) streaks, O(R −1 ) streamwise rolls and a weak sinusoidal wave that develops a critical layer, for large Reynolds number R. Higher harmonics become negligible. These unstable lower branch states appear to have a single unstable eigenvalue at all Reynolds numbers. These results suggest that the lower branch coherent states control transition to turbulence and that they may be promising targets for new turbulence prevention strategies.

Exact coherent states in no-slip plane Poiseuille flow are calculated by homotopy from free-slip to no-slip boundary conditions. These coherent states are unstable travelling waves. They consist of wavy low-speed streaks flanked by staggered streamwise vortices closely resembling the asymmetric coherent structures observed in the near-wall region of turbulent flows. The travelling waves arise from a saddle-node bifurcation at a sub-turbulent Reynolds number with wall-normal, spanwise and streamwise dimensions smaller than but comparable to 50+, 100+ and 250+, respectively. These coherent solutions come in pairs with distinct structure and instabilities. There is a three-dimensional continuum of such exact coherent states.