Free-surface measurements in a cylindrical container using synthetic schlieren with extended paraxial range

In this study, the free-surface synthetic schlieren (FS-SS) method, as proposed by Moisy et al. (Exp. Fluids 46 (2009) 1021), is extended to allow a wider application of the method. The reliance on small-angle paraxial approximation is removed, which allows for shorter distances of the camera to the free surface. Also, the presence of walls and non-constant depth of the liquid layer are accounted for. The difference between the original and modified approach is evaluated using a ray tracing approach. The extension of the method is demonstrated with measurements of the water surface in a cylinder for both rotating and non-rotating cases, i.e., surface waves due to impact, a surface depression due to a spin-up vortex, and the parabolic free surface in a rotating cylinder. The measured variations in surface elevation are as small as $O$ (10  $\mu$ m) with a range in paraxial angle up to ±0.56 rad ( ${32.3}^{\circ }$ ).

The online version contains supplementary material available at 10.1007/s00348-026-04240-z.