Mode Content of Piston-generated Initial Condition
One desirable initial condition for RM experiments is the sine wave
since there exist analytical models to calculate the amplitude
growth rate in the linear and nonlinear regimes. Some of these
theories may also be used on a multi-mode initial condition by using
superposition of modes to calculate growth. The modal
amplitudes will grow at different rates with the higher modes
(shorter wavelengths) saturating earliest. Therefore, for comparing
experimental growth rates with bubble- or spike-growth models, it is
important to have a dominant fundamental mode even if it is not a
perfect sine wave.
The He-SF6 interface generated with the pistons resembles a
curtate cycloid, or trochoid. The parametric equations for a
cosine and cycloid are shown below and the condition that a>b
means the cycloid is "curtate".
One wavelength of the initial condition is shown below. The
wavelength does not exhibit pure one-fold symmetry. Also, the
starting and ending points are not at the same exact locate, and
therefore, the average of these two points and the lowest point on the
trough are used to determine a peak-to-peak amplitude of 2&eta0.
An edge detection algorithm is performed on the experimental image to determine
the wave and an fft on that wave determines the mode content with only the first 10
being shown in the bar chart. Visually, the cycloid appears to come quite close to
the measured interface, and a quantitative comparison may be made by looking at the
modal content in the bar chart. Thus, this initial condition is much better
described as a cycloid than a cosine as it contains content in modes 2-4 that is close the
experimentally measured modes. Also, the fft shows the interface does contain
a fundamental dominant mode and is therefore suitable for comparison with many of
the analytic theories.