Giovanna Vittori

Coherent structures in oscillatory boundary layers

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.

1DV bottom boundary layer modeling under combined wave and current: turbulent separation and phase lag effects

On the basis of the Wilcox [1992] transitional k-ω turbulence model, we propose a new k-ω turbulence model for one-dimension vertical (1DV) oscillating bottom boundary layer in which a separation condition under a strong, adverse pressure gradient has been introduced and the diffusion and transition constants have been modified. This new turbulence model agrees better than the Wilcox original model with both a direct numerical simulation (DNS) of a pure oscillatory flow over a smooth bottom in the intermittently turbulent regime and with experimental data from Jensen et al. [1989] , who attained the fully turbulent regime for pure oscillatory flows. The new turbulence model is also found to agree better than the original one with experimental data of an oscillatory flow with current over a rough bottom by Dohmen-Janssen [1999] . In particular, the proposed model reproduces the secondary humps in the Reynolds stresses during the decelerating part of the wave cycle and the vertical phase lagging of the Reynolds stresses, two shortcomings of all previous modeling attempts. In addition, the model predicts suspension ejection events in the decelerating part of the wave cycle when it is coupled with a sediment concentration equation. Concentration measurements in the sheet flow layer give indication that these suspension ejection events do occur in practice.

Sand ripples under sea waves Part 3. Brick-pattern ripple formation

An oscillatory flow over a cohesionless bottom can produce regular three-dimensional bedforms known as brick-pattern ripples characterized by crests perpendicular to the direction of fluid oscillations joined by equally spaced bridges shifted by half a wavelength between adjacent sequences (a photo of brick-pattern ripples is shown in Sleath 1984, p. 141). In the present paper brick-pattern ripple formation is explained on the basis of a weakly nonlinear stability analysis of a flat cohesionless bottom subject to an oscillatory flow in which three-dimensional perturbations are considered.

Turbulent boundary layer under a solitary wave

The boundary layer generated by the propagation of a solitary wave is investigated by means of direct numerical simulations of continuity and Navier-Stokes equations. The obtained results show that, for small wave amplitudes, the flow regime is laminar. Turbulence appears when the wave amplitude becomes larger than a critical value which depends on the ratio between the boundary-layer thickness and the water depth. Moreover, turbulence is generated only during the decelerating phase, or conversely, turbulence is present only behind the wave crest. Even though the horizontal velocity component far from the bed always moves in the direction of wave propagation, the fluid particle velocity near the bottom reverses direction as the irrotational velocity decelerates. The strength and length of time of flow reversal are affected by turbulence appearance. Also the bed shear stress feels the effects of turbulence presence.

Migrating sea ripples

Ripple formation under sea waves is investigated by means of a linear stability analysis of a flat sandy bottom subject to the viscous flow which is present in the boundary layer at the bottom of propagating sea waves. Nonlinear terms in the momentum equation are retained to account for the presence of a steady drift. Hence the work by Blondeaux is extended by considering steeper waves and/or less deep waters. Second order effects in the sea wave steepness are found to cause neither destabilizing nor stabilizing effects on the process of ripple formation. However, because of the presence of a steady velocity component in the direction of wave propagation, ripples are found to migrate at a constant rate which is predicted as function of sediment and wave characteristics. The analysis assumes the flow regime in the bottom boundary layer to be laminar and the results are significant for ripples at the initial stage of their formation or for mature ripples of small amplitude (rolling-grain ripples). A comparison of the theoretical findings with laboratory experiments supports the reliability of the approach and of the theoretical results.

A route to chaos in an oscillatory flow : Feigenbaum scenario

The transition process which leads the oscillatory flow over a wavy wall from a periodic behavior to chaos is studied by means of the numerical algorithm described by Blondeaux and Vittori [J. Fluid Mech. 226, 257 (1991)]. By increasing the Reynolds number, it has been found that the flow experiences an infinite sequence of period doublings (pitchfork bifurcations) which take place at successive critical values. These critical values of the Reynolds number accumulate to a finite limit with the Feigenbaum rate of convergence. For Reynolds numbers larger than the above limit a chaotic flow is detected.

Quasiperiodicity and phase locking route to chaos in the 2‐D oscillatory flow around a circular cylinder

The two‐dimensional oscillatory flow around a circular cylinder is analyzed by means of the numerical approach described in Justesen [J. Fluid Mech. 222, 157 (1991)]. For a fixed value of the ratio between the Stokes viscous thickness and the radius of the cylinder section, when the Reynolds number assumes low values, the flow is periodic and symmetric with respect to an axis aligned with the flow direction and crossing the axis of the cylinder. An increase in the Reynolds number beyond a first critical value causes the flow to bifurcate: the velocity field loses its spatial symmetry even though it retains its time periodicity. When the Reynolds number is larger than a second critical value, a new frequency appears in the flow. This new frequency, which is much smaller than the frequency of the basic flow, increases for increasing values of the Reynolds number till a phase locking takes place. A further increase in the Reynolds number leads the flow to a chaotic status. The ‘‘quasiperiodicity and phase locking’’ route to chaos can be recognized.

Coherent structures in an oscillatory separated flow: numerical experiments

Numerical experiments are performed to investigate the oscillatory flow over a two-dimensional wavy wall characterized by a large amplitude, such as to induce flow separation. Even though the Reynolds number is moderate, a three-dimensional turbulent flow is observed. The turbulence dynamics is characterized by the presence of coherent ribs superimposed on the main spanwise vortices generated by the roll-up of the free vortex sheets shed at the crests of the wall waviness. The ribs are formed by the stretching of vorticity patches which are generated by the instability of the two-dimensional flow at two different locations. The first are the saddle points of the flow field created, far from the wall, by the vortex pairs generated every half-cycle. The second are the saddle points created, close to the upstream side of the wavy wall, by the combined action of the free-stream flow and of the vortex structures shed by the ripple crests. Later, the ribs wrap around the main spanwise vortices and cause the distortion of these vortices and the alignement of the vortex lines with the free-stream flow, thus inducing large contributions to the coherent helicity. Simultaneously, regions of high dissipation appear which tend to be separated from those characterized by large values of helicity.

Sea waves and mass transport on a sloping beach

The steady streaming induced by a sea wave shoaling on a sloping beach and partly reflected at the coastline is determined in the region seaward of the breaker line. Shallow waters and waves of small amplitude are considered. Moreover, the Reynolds number is assumed to be large but still within the laminar regime and the flow domain is split into a bottom boundary layer and a core region. For an incoming wave which is fully absorbed at the coast the solution shows that close to the bottom the steady streaming is onshore directed even though the depth–averaged value represents an offshore directed flow. Moreover, the vertical velocity distribution depends on the ratio between the wave amplitude a* and the thickness δ* of the bottom boundary layer. For a fully reflected wave, steady recirculation cells are induced, the form and strength of which depend on the ratio a*/δ*. A complex flow is generated for reflection coefficients falling between 0 and 1.

Long bed waves in tidal seas: an idealized model

An idealized model is proposed to explain the appearance of the long bed waves that have been recently observed in shallow tidal seas. The model assumes that these bedforms grow due to tide-topography interaction. The water motion is described by means of the depth-averaged shallow water equations and the bottom evolution is governed by conservation of sediment mass. The sediment transport formulation includes a critical bottom stress below which no sediment moves. Also, anisotropic sediment transport, due to local bottom slopes in the longitudinal and transverse directions, is taken into account. A linear stability analysis of the flat bottom configuration reveals that different bottom patterns can emerge. In accordance with previous analyses, for strong tidal currents, the fastest growing modes are sand banks. However, if the tidal currents are elliptical and the maximum bottom stress is just above its threshold value for the initiation of sediment motion, the model shows the presence of further growing modes which resemble the long bed waves observed in the field