nondimensionalization

Nondimensionalization is the process of transforming physical equations with dimensional variables into a dimensionless form by rescaling variables with characteristic units. By removing units, the equations expose the relative importance of different physical effects through dimensionless groups and facilitate comparison across systems with different scales.

Procedure: choose characteristic scales for length L, velocity U, time T, temperature ΔT, etc. Define dimensionless variables x* = x/L, t* = t/T, u* = u/U, T* = (T − T_ref)/ΔT, p* = p/(ρ U^2) (for fluids). Substitute into governing equations and simplify. Collect constants into dimensionless groups such as the Reynolds number Re = ρ U L/μ, Prandtl number Pr = μ c_p/k, etc. The resulting equations depend only on these groups, not on measurement units.

Example: In incompressible Navier–Stokes, with density ρ, viscosity μ, scales L and U, the dimensionless form is ∂u*/∂t* + (u*·∇*)u* = −∇*p* + (1/Re) ∇*^2 u*, ∇*·u* = 0, with Re = ρ U L / μ. For the energy equation, ∂T*/∂t* + u*·∇*T* = (1/(Re Pr)) ∇*^2 T*, where Pr = μ c_p / k, and T* is scaled by ΔT.

Applications and benefits: nondimensionalization is widely used in fluid dynamics, heat and mass transfer, combustion, and geophysical flows. It reduces parameters, reveals similarity across systems, and guides scaling for experiments and numerical simulations, while aiding asymptotic analyses.

Notes: choosing scales is problem-dependent; nondimensional equations reflect assumptions and approximations. Dimensionless groups aid interpretation but are not universal prescriptions, and results should be translated back to physical units for application.