Temporal length-scale cascade and expansion rate on planar liquid jet instability

Using the local radius of curvature of the surface and the local transverse dimension of the two-phase (i.e., spray) domain as length scales, we obtained two PDFs over a wide range of length-scales at different times and for different Reynolds and Weber ($We$) numbers. The PDFs were developed via post-processing of DNS Navier-Stokes results for a 3D planar liquid sheet segment with level-set and Volume-of-Fluid surface tracking, giving better statistical data for the length scales compared to the former methods. The radius PDF shows that, with increasing $We$, the average radius of curvature decreases, number of small droplets increases, and cascade occurs at a faster rate. In time, the mean of the radius PDF decreases while the rms increases. The other PDF represents the spray expansion in a more realistic and meaningful form, showing that the spray angle is larger at higher $We$ and density-ratios. Both the mean and the rms of the spray-size PDF increase with time. The PDFs also track the transitions between symmetric and anti-symmetric modes.