Modified free-surface synthetic Schlieren method to adjust measurement sensitivity in high-strain waves

This research revisits the free-surface synthetic Schlieren technique (FS-SS) (Moisy et al. in Exp Fluids 46(6):1021-1036, 2009) for the topographical measurement of the free surface of a liquid. In contrast to the aforementioned method, which utilizes the image refraction of a random dot pattern through a free surface to determine the surface gradients, this present study mathematically derives a method where the pattern may take an arbitrary three-dimensional shape. This is shown to be theoretically valid under a small pattern slope approximation. Our method is then verified against the previously mentioned, flat-pattern, free-surface synthetic Schlieren technique by resolving the free-surface elevation of plane waves across a channel, showing similar results in constant strain conditions, with improved results in variable strain conditions, particularly in cases where there is a very large difference in strain across the channel. The validation test cases investigated include a rectangular channel containing a transparent liquid with a random dot pattern placed below at a constant angle and a pattern placed on top of a cosine-shaped profile. Both of these setups are validated against the classical FS-SS technique involving a flat pattern. The new method involving an arbitrarily shaped pattern proposed here may increase the resolution in low-amplitude regions by increasing the surface-pattern distance below these regions and correspondingly reducing the sensitivity in high-strain regions by decreasing the surface-pattern distance. Geometries shown to produce advantageous results in waves that include both regions of very high strains and regions of very low amplitudes are explored, resolving the wave in both regions simultaneously. This shows promise in resolving multi-scale surface waves in highly viscous liquids, which may include very high-amplitude regions quickly followed by very low-amplitude regions due to damping effects.