Batchelor Scaling in Fast-Flowing Soap Films

The dynamics of a passive scalar such as a dye in the far dissipative range of fluid turbulence is a central problem in nonlinear physics. An important prediction for this problem was made by Batchelor over 40 years ago and is known as Batchelor's scaling law. We here present strong evidence in favor of this law for the thickness fluctuations in the flow of a soap film past a flat plate. The results also capture the dissipative range of the scalar which turns out to have universal features. The probability density function of the scalar increments and their structure functions come out in nice agreement with theoretical predictions.

Figure 1

(a) A schematic of the setup. The flat plate was inserted perpendicularly to the film plane and is indicated by the thick solid line. The film is replenished by a micropump pumping water from a reservoir of soap water placed at the bottom of the film to the top of the channel. Photographs of the wake in reflection from a white light source: (b) full area, (c) a zoom on a $1{\mathrm{c}\mathrm{m}}^2$ region, and (d) monochromatic light image [same area as (c)]. The color bar indicates film thickness in $\mu \mathrm{m}$.

Figure 2

Velocity power spectra versus the frequency. Inset: PDFs of the velocity fluctuations.

Figure 3

Thickness fluctuation spectrum from the temporal measurements. The left arrow indicates the position of $f_K$ and the right arrow indicates the position of the frequency corresponding to $r_B$. Upper inset: probability density functions of the thickness fluctuations relative to the mean thickness which is $1.5\mu \mathrm{m}$. Lower inset: $E_h(k)$ from spatial measurements using reflection from a sodium lamp. The left and right arrows indicate the $k$ values corresponding to $r_K$ and $r_B$, respectively.

Figure 4

(a) Second- and fourth-order structure function of the scalar increments in a linear-log plot. (b) PDFs of the scalar increments for different spatial scales $r$. The PDFs were translated upwards from each other for better visibility.