Betz invariants and generalization of vorticity moment invariants

The widely used Betz invariants can be categorized as invariants of vortici ty moments or impulses in plane motion for mostly inviscid vortex flow. The se invariants have served as a foundation of vortex roll-up process analysi s as well as benchmarks to estimate numerical simulation errors for vortex How: The analysis starts from a more general integral divergence relation i n rational mechanics to extend the analysis For both inviscid and viscous v ortex Row cases, with or without finite boundaries, Conditions for the exis tence of the Betz invariants and their extensions to include viscous effect s are discussed. These extensions can be used as analytical means for theor etical modeling of complicated vortex systems, especially in Rows with visc ous effects and finite boundaries. The goal of this study is to clearly est ablish when the Betz invariants and the generalized vorticity moment invari ants are applicable. The approach uses rigorous mathematical analysis, and the physical interpretations of the results are discussed, A trailing vorte x pair is used as an example to illustrate the applications of the theoreti cal study.