Dimensionslose

Dimensionslose, or dimensionless quantities, describe values without physical units. They arise when quantities with the same dimension are divided, or when a system is nondimensionalized using characteristic scales. Such numbers are invariant under changes of unit systems and are central to comparing different experiments, models, or simulations.

Dimensionless numbers are often constructed through ratios or the Buckingham Pi theorem, which reduces a problem to a set of nondimensional groups. They reveal the fundamental scales governing a process and enable data from diverse conditions to be collapsed onto a single curve. In practice, nondimensionalization simplifies mathematical models and improves numerical stability.

Common examples include the Reynolds number, Re = ρ v L / μ, which characterizes flow regime in fluid dynamics; the Mach number, Ma = v / a, describing compressibility effects; the Prandtl number, Pr = μ cp / k, linking momentum and thermal diffusion; the Nusselt number, Nu, and the Biot number, Bi = hL / k, used in heat transfer. In addition, geometric ratios like the aspect ratio are dimensionless, and certain fundamental constants, such as the fine-structure constant α, are inherently dimensionless. Angles, expressed in radians or steradians, are also treated as dimensionless in many physical theories.

Applications of dimensionless analysis include scaling, model verification, and the comparison of systems across different sizes and units. They also aid in interpreting experimental results and guiding the development of universal laws by highlighting similarity between seemingly disparate situations.