Numerical simulations of Navier–Stokes equations are performed to study the flow originated by an oscillating pressure gradient close to a wall characterized by small imperfections. The scenario of transition from the laminar to the turbulent regime is investigated and the results are interpreted in the light of existing analytical theories. The ‘disturbed-laminar’ and the ‘intermittently turbulent’ regimes detected experimentally are reproduced by the present simulations. Moreover it is found that imperfections of the wall are of fundamental importance in causing the growth of two-dimensional disturbances which in turn trigger turbulence in the Stokes boundary layer. Finally, in the intermittently turbulent regime, a description is given of the temporal development of turbulence characteristics.

The dynamics of the vortex structures appearing in an oscillatory boundary layer (Stokes boundary layer), when the flow departs from the laminar regime, is investigated by means of flow visualizations and a quantitative analysis of the velocity and vorticity fields. The data are obtained by means of direct numerical simulations of the Navier–Stokes and continuity equations. The wall is flat but characterized by small imperfections. The analysis is aimed at identifying points in common and differences between wall turbulence in unsteady flows and the well-investigated turbulence structure in the steady case. As in Jimenez & Moin (1991), the goal is to isolate the basic flow unit and to study its morphology and dynamics. Therefore, the computational domain is kept as small as possible.The elementary process which maintains turbulence in oscillatory boundary layers is found to be similar to that of steady flows. Indeed, when turbulence is generated, a sequence of events similar to those observed in steady boundary layers is observed. However, these events do not occur randomly in time but with a repetition time scale which is about half the period of fluid oscillations. At the end of the accelerating phases of the cycle, low-speed streaks appear close to the wall. During the early part of the decelerating phases the strength of the low-speed streaks grows. Then the streaks twist, oscillate and eventually break, originating small-scale vortices. Far from the wall, the analysis of the vorticity field has revealed the existence of a sequence of streamwise vortices of alternating circulation pumping low-speed fluid far from the wall as suggested by Sendstad & Moin (1992) for steady flows. The vortex structures observed far from the wall disappear when too small a computational domain is used, even though turbulence is self-sustaining. The present results suggest that the streak instability mechanism is the dominant mechanism generating and maintaining turbulence; no evidence of the well-known parent vortex structures spawning offspring vortices is found. Although wall imperfections are necessary to trigger transition to turbulence, the characteristics of the coherent vortex structures, for example the spacing of the low-speed streaks, are found to be independent of wall imperfections.

The process which leads to the appearance of three-dimensional vortex structures in the oscillatory flow over two-dimensional ripples is investigated by means of direct numerical simulations of Navier–Stokes and continuity equations. The results by Hara & Mei (1990a), who considered ripples of small amplitude or weak fluid oscillations, are extended by considering ripples of larger amplitude and stronger flows respectively. Nonlinear effects, which were ignored in the analysis carried out by Hara & Mei (1990a), are found either to have a destabilizing effect or to delay the appearance of three-dimensional flow patterns, depending on the values of the parameters. An attempt to simulate the flow over actual ripples is made for moderate values of the Reynolds number. In this case the instability of the basic two-dimensional flow with respect to transverse perturbations makes the free shear layer generated by boundary layer separation become wavy as it leaves the ripple crest. Then the amplitude of the waviness increases and eventually complex three-dimensional vortex structures appear which are ejected in the irrotational region. Sometimes the formation of mushroom vortices is observed.

In the present paper we determine the oscillatory flow generated by surface gravity waves near a sea bottom covered with large-amplitude ripples. The vorticity equation and Poisson equation for the stream function are solved by means of a numerical approach based on spectral methods and finite-difference approximations. In order to test the numerical algorithm and in particular the numerical scheme used to generate vorticity along the ripple profile, we also perform an asymptotic analysis, which holds as the timettends to zero. The main features of the time development of vorticity are analysed and particular attention is paid to the dynamics of the large vortices generated by flow separation at the ripple crests and along the ripple profile. Some of the results obtained by Longuet-Higgins (1981) are recovered; in particular, the present results show a vortex pair shed from the ripple crest every half-cycle. The determination of flow separation along the ripple profile induced by the pressure gradient and the inclusion of viscous effects allows us to obtain accurate quantitative results and detect some important phenomena never observed before.In particular it is shown that: (i) Whenever a vortex structure moves towards the bottom, a secondary vortex is generated near the ripple profile, which interacts with the primary vortex and causes it to move away from the bottom, (ii) Depending on the values of the parameters, the time development of the free shear layer shed from the ripple crest may produce two or even more vortex structures, (iii) Occasionally vortices generated previously may coalesce with the free shear layer shed from the ripple crest, generating a unique vortex structure.

In the present paper we formulate a theory to predict the time development of sand ripples characterized by small but finite amplitude under the action of surface gravity waves. The theory is based on a weakly nonlinear stability analysis of a flat sandy bottom subject to viscous oscillatory flow. The parameters of the problem (namely the Reynolds number of the flow and the Reynolds and Froude numbers of sediments) are assumed to fall within a neighbourhood of the critical conditions determined in Blondeaux (1990). The analysis can predict the actual ripple height, wavelength and profile when flow separation is absent, i.e. for the case of rolling-grain ripples. Assuming Sleath's (1984) criterion for separation, the values of the relevant parameters at which transition from rolling-grain ripples to vortex ripples occurs are predicted. A comparison between theoretical findings and experimental data supports the validity of the present theory.