Contour analysis

A contour detection method was developed to enable us to capture the different waves and large-scale behaviour of the flow. By examining a light level contour, we can estimate the different wavenumbers. To this end, we first have to capture the adequate light level. Next, we can find the arc-length of the contour and compute the fft. Finally, we consider the time evolution of the wavenumbers. The following paragraphs elaborate the different steps of this method.

First, the image is cleaned to avoid capturing any spurious effect. To exclude contours around the edge of the tank and because the dynamics take place in its centre, we create a circular mask, whose diameter is defined by the size of the image. Next, to remove small-scale fluctuations that we consider noise, we apply a smoothing routine, which will be presented in the following paragraph. We then choose a light level (see below for more details) and obtain the contour at this light level. Note that the contour is obtained

in a grid four times finer than the original image, which gives a very precise and neat contour, and using the contour detection algorithm developed byDritschel and Ambaum

[1997]. Next, we compute the arc-length of the contour between contour points. Then, we redistribute the contour points so that each point is separated from the next one by the same arc length. Thus, for each point on the contour, we obtain its position(x, y) as a function of arc lengths. Finally, we compute the 1D ffts ofx(s)andy(s). By summing the square of the two spectra, we obtain the power spectrum of the contour shape.

Figure 4.5: Contours obtained from experiment 1, at time t = 405s for the light level

l = 32882for four different levels of smoothing. Black line: no smoothing, blue line:

ks= 5smoothing iterations, red line:ks= 10smoothing iterations, green line: ks= 30

smoothing iterations. Left panel: whole contour, right panel: zoom on a specific region.

The smoothing routine is based on a diffusive process. Images can display many small variations from one pixel to another, due, for example, to the presence of a particle, or any other perturbation on the laser beam. In order to smooth the data, a diffusive filter is applied of the form data(x, y) = 0.5data(x, y) + 0.25(data(x −1, y) +data(x+ 1, y)) as equally applied to the y direction. Different numbers of iterations ks of this

diffusive process were tested. Figure4.5exhibits contours obtained from experiment 1 at timet = 405s for the contour levell = 32882 for different smoothing treatments. The black line corresponds to no smoothing, the blue line toks = 5iterations, the red line

to ks = 10 iterations, and the green line to ks = 30 iterations. The left panel shows

the whole contour while the right panel zooms into the waves with the wavelengths of greatest interest. We can see that the smoothing process represented by the black line captures very small scales. These small scales are in part due to very small fluctuations from one pixel to another. Typically, at the rim of the tank where the laser enters it, small perturbations such as bubbles can create small waves on the contours. Conversely, the smoothing process represented by the green line largely inhibits the waves we are

interested in. The interesting waves are captured best usingks = 5toks = 10smoothing

iterations.

Figure 4.6: Log profile of the power spectrum around a level contour equal to 32884, at time150s, for various smoothing iterations. Black line: no smoothing, blue line: ks =

5 iterations, red line: ks = 10 smoothing iterations, green line: ks = 30 smoothing

iterations.

Figure 4.6 shows the logarithm power spectrum of a light level contour equal to

l = 32882 at time105s for different smoothing iterations, for experiment 1. Note that for small wavenumbers, the results are very similar for most of the different numbers of smoothing iterations. For higher wavenumbers however, the smoothing acts as a low-pass filter. The more we iterate the smoothing process, the more we damp the high wavenum- bers. In the following analyses, we chose to use ks = 10 iterations (red line). For ks

smaller than five, the contour exhibits too many fluctuations. We chose ten iterations rather than five, as this results in a clearer profile.

To find the relevant light level which will capture the front, we plot the area covered by this level in the image (i.e. a probability density function not scaled between 0and

1). Figure 4.7 shows the area covered by each level. The interface is characterised by a region of a high positive gradient of the area covered by each contour. The first high positive gradient corresponds to the centre of the tank, while the most interesting contour levels are found in the second and third high positive gradients.

This method yields a precise spectrum of the wavenumbers around a specific contour. However, as the contour’s size changes with time, so too do the wavenumbers. This method is, therefore, used mainly for single images rather than time evolutions.

Figure 4.7: Area covered by each light level compared with light intensity levels. The black line exhibits the time averaged area covered by each level over 9000 images and the grey regions exhibit the time averaged area minus the area covered by each level variance over 9000 images.