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Faraday experiment

 

I will show plots of tex2html_wrap_inline4715, tex2html_wrap_inline4717, and tex2html_wrap_inline4719, for seven different driving amplitudes. I will compare the results from this analysis with the results presented in [1] (appendix C). The present analysis based on Kolmogorov's choice of which data should be counted, whereas our earlier analysis was heavily influenced by Richardsons view.

The implementation of the particle tracking algorithm we used for the data aquisition gives an over-representation of particles which are close by each other, but this should not give rise to any concern. Whether we take ten or thousand samples should the probability distribution of the velocities be identical.

Lengths are measured in mm and times are measured in ms, in the graphs presented in this and the following section.

  table1121
Table 5.1: Driving amplitudes and the corresponding reduced control parameter.

  figure1130
Figure 5.5: Left: tex2html_wrap_inline4715 as a function of R. Right: The gradients of the graphs in the plot on the left as a function of tex2html_wrap_inline4757.

Figure 5.2 shows tex2html_wrap_inline4715 as a function of R (on the left), and locally defined powers of R that fit the function (on the right). None of the curves have even something close to a constant gradient over a whole decade. The maximum gradient occurs at a particle separation equal to one wavelength (tex2html_wrap_inline4461).

  figure1147
Figure 5.3: tex2html_wrap_inline4767 as a function of R. This analysis is limited to look at relative particle tracks from the moment the two particles were closest to each other (and closer than tex2html_wrap_inline4771) and onwards.

The data plotted in figure 5.3 scales roughly as R to powers in the range 0.8 to 1. This is equivalent to the powers slightly below 2 you see for allmost a decade in figure 5.2. So even though we have different rules for which data should be analysed, we get similar results (just as Kolmogorov's turbulence theory fits Richardson's data quite well).

  figure1157
Figure: Left: tex2html_wrap_inline4717 as a function of R. Right: The gradients of the graphs in the plot on the left as a function of tex2html_wrap_inline4757.

I do not have an interpretation of tex2html_wrap_inline4787, but it shows a much nicer behaviour than tex2html_wrap_inline4767, so I would like to know if it is possible to relate it to existing turbulence theory.

Figure 5.4 shows tex2html_wrap_inline4717 as a function of R (on the left), and locally defined powers of R that fit the function (on the right).

tex2html_wrap_inline4787 is roughly proportional to tex2html_wrap_inline4799 for tex2html_wrap_inline4801.

  figure1177
Figure: Left: tex2html_wrap_inline4719 as a function of R. Right: The gradients of the graphs in the plot on the left as a function of tex2html_wrap_inline4757.

  figure1187
Figure 5.6: tex2html_wrap_inline4719 divided by tex2html_wrap_inline4811 as a function of R. This should be a horizontal line for wave turbulence.

Figure 5.5 shows tex2html_wrap_inline4719 as a function of R (on the left), and locally defined powers of R that fit the function (on the right). There is no distinct power law in these data.

Weak turbulence theory predicts that tex2html_wrap_inline4821 is proportional to tex2html_wrap_inline4823 (see definition of b in section 1.2.4), and thus that tex2html_wrap_inline4719 is proportional to tex2html_wrap_inline4811. Figure 5.6 shows the data divided by the predicted function. From tex2html_wrap_inline4831 and up to the maximum value for R does the weak turbulence prediction fit with the data.

The factor needed to fit the data to the function depends on the energy dissipation in the system (please see [14] for the details).

  figure1216
Figure 5.7: The upper graph shows the number of data points per mm in the old analysis, and the lower graph shows the same data for the analyses made for this thesis. Both of the graphs are for a driving amplitude of tex2html_wrap_inline4837.

Figure 5.7 shows the densities of data points used for some of the graphs in the preceding figures. It is not a big surprise that there are more data available for the analyses made for this thesis, if you consider that I have not ruled any of the available data out beforehand.


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