Dynamic similarity, the dimensionless science

www.physicstoday.org Many experiments seem daunting at first glance, owing to the sheer number of physical variables they involve. To design an apparatus that circulates fluid, for instance, one must know how the flow is affected by pressure, by the apparatus’s dimensions, and by the fluid’s density and viscosity. Complicating matters, those parameters may be temperature and pressure dependent. Understanding the role of each parameter in such a high-dimensional space can be elusive or prohibitively time consuming. Dimensional analysis, a concept historically rooted in the field of fluid mechanics, can help to simplify such problems by reducing the number of system parameters. For example, in a fluid apparatus in which the flow is isothermal and incompressible, the number of relevant parameters can often be reduced to one. The rewards of such a reduction can be spectacular: It may allow a model the size of a children’s toy to yield insight into the dynamics of a jet airplane, or a fluid-filled cylinder the size of a garbage can to elucidate the behavior of a stellar interior. (See box 1 for a brief history of dimensional analysis.)