Reconstruction of sub-surface flows from surface manifestation

Sub-surface flows imprint distinct signatures on the water surface. These patterns include, but are not limited to, scars, dimples, and boils, which correspond to horizontal vortices underneath the surface, surface-connected vortex tubes, and upwelling, respectively. Inferring the associated sub-surface flows from surface measurements is of great importance to many applications. In this study, we use the adjoint method to reconstruct the velocity field using the surface data. We employ the coupled level set and volume-of-fluid (CLSVOF) method to simulate two-phase flows. Subsequently, we use observation of the volume-of-fluid field to reconstruct the velocity field on both the air and water sides. We investigate various test cases, ranging from low-steepness waves to vortex pair rising. Results show that reconstructing the subsurface velocity field is more accurate than reconstructing the above-surface velocity field due to the higher sensitivity of the surface to water flow structures. Furthermore, for irrotational flow fields, consistent with potential flow theory, surface information can fully reconstruct the sub-surface velocity field. In contrast, for the rotational vortex pair rising case, the sub-surface velocity derived from the surface manifestation is not unique. While the sub-surface velocity field near the surface is reconstructed accurately, the vortex core and vortex strength are not precisely captured. In such cases, incorporating prior knowledge of the sub-surface structures can help reconstruct the key characteristics of the sub-surface flow more satisfactorily.