Options for Dynamic Similarity of Interacting Fluids, Elastic Solids and Rigid Bodies Undergoing Large Motions

Motivated by a design of a vertical axis wind turbine, we present a theory of dynamical similarity for mechanical systems consisting of interacting elastic solids, rigid bodies and incompressible fluids. Throughout, we focus on the geometrically nonlinear case. We approach the analysis by analyzing the equations of motion: we ask that a change of variables take these equations and mutual boundary conditions to themselves, while allowing a rescaling of space and time. While the disparity between the Eulerian and Lagrangian descriptions might seem to limit the possibilities, we find numerous cases that apparently have not been identified, especially for stiff nonlinear elastic materials (defined below). The results appear to be particularly adapted to structures made with origami design methods, where the tiles are allowed to deform isometrically. We collect the results in tables and discuss some particular numerical examples.