Boundary layer turbulence

This is illustrated by figure (a–d) that shows x–z slices of the ∂w/∂z field in a frame moving with free-stream velocity at four different phases of the tidal cycle. The upper panels in the figure show the far-field waves while the middle panel shows turbulence as well as the near-field waves adjacent to the generation region. Clearly, the slope of the phase lines depends on tidal phase and the slope differs between far- and near-fields. The bottom panel, which shows mean velocity profiles at the corresponding times with black showing values positive with respect to the free stream and grey showing negative values, helps to explain the observed phase lines. Although, it is difficult to exactly demarcate the wave source, the generation region is bounded from above by the location of ∂θ(z, t )/∂z=0.1, the top of the mixed layer. The snap in figure (a) corresponds to a late decelerating phase (φ∼−5◦) where a new boundary layer with reverse flow in the bottom relative to the free stream has been formed. The flow velocity, relative to the free stream, is positive in the source region as shown in the bottom panel of figure (a) leading to phase lines inclined to the front as shown in the middle panel of figure (a). These forward-inclined waves do not have enough time to propagate into the outer region and the backward-inclined phase lines in the outer region, shown in the upper panel of figure (a), correspond to internal waves generated during the early accelerating stage of the previous cycle, for example, those in the middle panel of figure (c–d). When the flow progresses in time, a near-bottom region of negative velocity with respect to the free stream, similar to the conventional steady boundary layer, progressively develops. Figure shows φ =90◦ where, as shown by the middle panel, waves are generated from the source with no preferred inclination of the phase lines, see middle panel, but the outer region shown in the top panel exhibits forward-inclined waves that were generated previously, for example at φ =−5◦ shown in figure.

Now, examination of the velocity profile at φ =90◦ reveals that most of the mixed layer has negative relative velocity compared to a small portion with positive relative velocity. Therefore, one would expect generation of backward waves from the source region and, indeed, such phase lines were observed in the steady stratified Ekman boundary layer by Taylor & Sarkar (2007) where the streamwise velocity profile has a shape similar to that seen here at φ =90◦. However, the history of the mean flow is important in the oscillating case; forward phase lines emitted by the predominantly positive velocity at an earlier nearby phase remain adjacent to the boundary layer at φ =0◦ if cg,z is sufficiently small and, consequently, both forward and backward phase lines are observed in the middle panel of figure (b). Thus, owing to history effects, arriving at a conclusion based on steady currents that, at φ =90◦ in the present oscillating flow, the phase lines would tilt backward is clearly erroneous. During the following deceleration stage, the source is dominated by fluid with negative relative velocity giving rise to waves with phase lines inclined towards the back (middle panel of figure c) while the outer region still has the forward phase lines of internal waves generated in the previous acceleration stage. In figure (d), the phase of −90◦ corresponding to negative freestream velocity is shown. Here, similar to the phase of 90◦ shown in figure (b), there is no clear direction of phase lines adjacent to the boundary layer. However, the phase lines in the far field tilt backward.

Ref. B. Gayen and S. Sarkar and J. R. Taylor (2010) LES of a stratified boundary layer under an oscillatory current.
J. Fluid Mechanics, 643, p. 233–266.