8.2 Spatial distribution of vorticity
8.2.1 Instantaneous vorticity, ω y
In-to-plane vorticity is easily derived from the measured velocity flow fields. Considering a two-dimensional fluid flow in thex−z plane, whose velocity field is confined to a plane, the vorticity component, ωy, which points in the orthogonaly-direction reduces to (Sou &Yeh [12] :
ωy= dui
dz −dwi
dx (8.2)
where the indexidenotes instantaneous velocities.
Instantaneous vorticity fields were computed from derivatives of the instantaneous horizontal, ui, and vertical velocity,wi components, using the central difference method (Sou &Yeh[12]; Lee &Lee[195]) as follows :
ωy =ui(i, j+ 1)−ui(i, j−1)
2∆z −wi(i+ 1, j)−wi(i−1, j)
2∆x (8.3)
where ∆xand ∆z are thexand z-grid spacing, respectively andi andj are integers. Thus the instan- taneous vorticity normal to the plane of the light sheet was obtained at each grid point by calculating the circulation. Figure 8.1 shows the grid points in a velocity mesh used for the calculation of vorticity at a particular point. The four yellow-shaded grid points were used to numerically estimate the vorticity at point (i,j). This vorticity was computed by the circulation around squares of 5.0 mm×6.5 mm. To fully resolve the full turbulent vorticity field and reveal smallest scale structures of the flow may require a finer resolution.
Figure 8.2 shows the contour plots of instantaneous vorticity that reveals the evolution of the different vortex structures and vorticity fields. These plots were obtained from the instantaneous velocity fields presented in Figures 6.3 - 6.4 which are instantaneous velocity fields for those images shown in Figure 6.2.
Figure 8.1: Velocity mesh grid used for the calculation of vorticity at point (i,j).
These six snapshots of vorticity field are here presented to best illustrate spatio-temporal evolution of near-surface eddies during the passage of the wave crest. The colour bar shows the magnitude and direction of the vorticity. Positive vorticity indicates motion in clockwise rotation and the direction is into the plane of the figure, while negative vorticity indicates anticlockwise rotation with direction out of the plane of the figure. Positive vorticity has been conveniently taken to point in the positive y direction. Red-filled contours indicate clockwise or positive vorticity whereas the blue-filled contours represent counterclockwise or negative vorticity.
The presence of high-frequency vorticity patches near the water surface, shows that wave breaking that occurs at the free surface is the major source of vorticity in the flow. It can be observed that the instantaneous vorticity distribution is extremely patchy, with isolated pockets of high and low vorticity.
These patches begin to appear near the toe of the breaker as observed at phase 0.00. In between the patches and below the SWL are large expanses with nearly zero vorticity. Results also show the presence of pairs of counter-rotating fluid patches with peak vorticity of magnitude 150s−1. These patches have only been shown to exist, but it is not clear the mechanism by which they are set up. However these high vorticity eddies dissipate the remaining wave energy. It is the interactions between these adjacent counter-rotating eddies that produce a wake of complex vortex distribution behind the wave crest. These pairs have almost similar orientation in the flow. Bakewell &Lumley [196] andAubry et al.[197] used the proper orthogonal decomposition in the near wall region and determined that a pair of counter- rotating streamwise vortices contain the largest amount of energy. Ting [198], [199] used PIV to study instantaneous turbulent velocity fields associated with a broken solitary wave on a plane slope and also observed that large eddies were composed of two counter-rotating vortices. Peak values of instantaneous vorticity are observed to decrease as flow progresses.
As the flow progresses, patches of intense vorticity are observed to be generated at the free surface and are observed to diffuse to the bottom of the flume reaching the bed after the crest has passed (phases 0.20 and 0.25). This may be an indication that large eddies in the flow become unstable and degenerate into moderate-scale structure of the main turbulent motion. Eddies with anticlockwise vorticity are observed to disappear as a result of mixing leaving the fluid volume mainly filled with clockwise vortex structures.
According toLonguet-Higgins [200], vorticity is generated at the free surface with intensity proportional
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−10
−5 0 5 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
t/T = 0.00
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−10
−5 0 5 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
t/T = 0.05
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−10
−5 0 5 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
t/T = 0.10
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−10
−5 0 5 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
t/T = 0.15
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−10
−5 0 5 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
t/T = 0.20
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−10
−5 0 5 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
t/T = 0.25
(s−1)
−100 −50 0 50 100
Figure 8.2: Typical contour plots showing the phase evolution of instantaneous vorticity,ωy, for breaking wave images given in Figure 6.2.
to the tangential velocity and local curvature. Steepening of the wave naturally induces high curvature and consequently strong vorticity. The vortex structures deepen and diffuse towards the bottom as shown in phases 0.10 - 0.25. As pointed out byLongo [66] deepening is quite fast immediately after breaking and is slower at subsequent phases. Nadaoka et al. [201] observed that the large eddies are initially two-dimensional, but break down into three-dimensional eddies descending obliquely downward.
Saddle points near the crest are characterized by strips of strong obliquely descending positive and negative vorticity below them, which can be observed near the saddle points for Tt= 0.10 - 0.20. These eddies first observed at about x = -250 cm for phase 0.10 originate from the free surface and advect horizontally in the direction of wave propagation. They are an indication of strong mixing taking place there. It has been reported that two compact vortices of the same sign rotate around each other, while two vortices of opposite signs translate as a unitJimenez et al.[202]. As flow progresses, these patches of counter-rotating vorticity interfere with each other by viscous vorticity cancellation leaving dominantly positive vorticity in the flow which diminish in magnitude with a relatively uniform distribution behind the crest. This cancellation is the key mechanism for the decay of two-dimensional vorticity distributions.
While the energy moves to larger scales by amalgamation, the vortex debris being generated during shedding cancels with other structures of the opposite sign by viscous diffusion.
It is possible to make some rough estimate of the characteristics of the vortex structures in the flow, e.g.
the vortex core radius, vortex length and velocity of the vortex between frames as the structures travel downstream in the flow. These strips are estimated to have length scales that stretch up to 12 cm and a typical mean vortex core radius of about 1 cm. In the contour plots a large clockwise vortex structure of magnitude about 60s−1 is observed which is centered around (x, z) = (-250, -7) cm at phase 0.20, and is observed to develop into a vortex ring that has moved to position (x, z) = (-239, -7) cm in the plot of phase 0.25. Thus the vortex moves about 10 cm in 1.84 s so that roughly the vortex structures in the flow propagate at 6 cm/s. The wave speed is nearly twenty times the vortex speed, meaning that it would take the vortex twenty wave periods to move a wavelength. In agreement with observation by Kimmoun &Branger [43] the eddies associated with the flow under study are observed to be advected obliquely towards the bottom and moves slower than the wave crest. This type of plunging breaker generation produces a thin higher-velocity free-surface fluid layer that decelerates just prior to breaking, thus injecting a large amount of vorticity into the fluid bulk, entirely through a viscous process.
Figure 8.3 shows in one frame, the evolution of profiles of instantaneous vorticity with depth as flow evolves. From the flume bed up to the SWL, vorticity is about zero and increases to peak values of over 150 s−1 just above the SWL before dropping to negative vorticity values towards the top of the crest. In agreement with observations byMisra et al.[72], it is also observed that the variations of the instantaneous vorticity profiles with depth show well defined peaks, indicating coherent vortical motions in the shear layer.
Figure 8.4 shows instantaneous vorticity contours under the wave crests measured at five different cross- shore positions along the flume, corresponding to the five measurement stations. The position of the bed is different in each figure. While the size of counter-rotating vorticity patches decreases towards the shore,
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−15
−10
−5 0 5 10 15
Elevation, z, relative to SWL (cm)
Instantaneous vorticity, (s−1)
bed position phase = 0.00 phase = 0.05 phase = 0.10 phase = 0.15 phase = 0.20 phase = 0.25
Figure 8.3: Profiles of instantaneous vorticityωy as a function of depth, measured atx= -238 cm as flow progresses.
the intensity is observed to increase. This may again be caused by increased resolution used towards the shore, which enables the identification of small but equally intense vorticity patches in the flow. Both negative and positive vorticity of magnitude up to 100s−1 can be observed impinging the flume bed as flow progresses. Such high vorticity fluid elements are responsible for lifting sediments from the bed and transporting them. Statistics on the number of high vorticity patches impinging the bottom and their magnitudes can be determined and is vital information in sediment transport studies.