dimensional-analysis

name	dimensional-analysis
description	Dimensional analysis — Buckingham Pi theorem, dimensional homogeneity, similarity laws, scaling, key dimensionless groups (Re, Fr, Nu, Bi, We, Ma, St, Eu), model testing.
metadata	{"priority":7,"promptSignals":{"phrases":["dimensional analysis","Buckingham Pi","dimensionless","similarity","scaling","Reynolds number","Froude number","Nusselt number","model testing","Pi theorem"],"minScore":3}}

Dimensional Analysis — Complete Skill

Dimensional Homogeneity

Every physically meaningful equation must be dimensionally homogeneous. Primary dimensions (SI): M [kg], L [m], T [s], θ [K], I [A]

Check by substituting dimensions: Example: F = ma → [kg·m/s²] = [kg][m/s²] ✓ Fourier: q = -k dT/dx → [W/m²] = [W/mK][K/m] ✓

Buckingham Pi Theorem

Statement

If a physical phenomenon involves n variables and r independent primary dimensions: Number of dimensionless Pi groups = n - r

Procedure

1. List all n variables: variable list with units
2. Identify r = number of primary dimensions involved (typically r ≤ 5)
3. Choose r repeating variables (must span all dimensions; avoid choosing the dependent variable)
4. Form (n-r) Pi groups by combining repeating variables with each remaining variable
5. Make each Pi dimensionless by inspection (solve exponent equations)

Repeating variable selection rules:

• Must include all primary dimensions
• Must NOT include the dependent variable
• Avoid dimensionless quantities as repeating variables
• Common choices: (ρ, V, L), (μ, V, D), (k, V, L)

Example — Pipe Flow Pressure Drop

Variables: ΔP, ρ, V, D, μ, L, ε (roughness) n=7, dimensions: M, L, T → r=3 Pi groups = 7-3 = 4

Choose repeating: ρ, V, D Π₁ = ΔP/(ρV²) = Euler number Π₂ = μ/(ρVD) = 1/Re = 1/Reynolds number Π₃ = L/D = slenderness ratio Π₄ = ε/D = relative roughness

Result: ΔP/(ρV²) = f(Re, L/D, ε/D) → Darcy-Weisbach recovered!

Key Dimensionless Groups

Fluid Mechanics

Reynolds: Re = ρVL/μ = VL/ν Ratio: inertia / viscous forces Re < 2300: laminar pipe flow; Re > 4000: turbulent Re_critical (flat plate): ~5×10⁵

Froude: Fr = V/√(gL) Ratio: inertia / gravity forces Fr < 1: subcritical (tranquil); Fr > 1: supercritical (rapid) Critical for: open channel flow, ship hydrodynamics, hydraulic jumps

Mach: Ma = V/a = V/√(γRT) Ratio: flow velocity / speed of sound Ma < 0.3: incompressible; 0.3-0.8: subsonic compressible; 0.8-1.2: transonic; > 1.2: supersonic

Weber: We = ρV²L/σ Ratio: inertia / surface tension Important for: droplet formation, atomization, thin films, We > 12 → droplet breakup

Strouhal: St = fL/V Ratio: oscillation frequency × length / velocity Vortex shedding: St ≈ 0.2 (circular cylinder, Re=10³-10⁵)

Euler: Eu = ΔP/(ρV²) — pressure / dynamic pressure Cavitation number: σ_ca = (P - P_v)/(½ρV²) — low → cavitation risk Drag coefficient: C_D = F_D/(½ρV²A)

Heat Transfer

Nusselt: Nu = hL/k_fluid Ratio: convective / conductive heat transfer Nu = f(Re, Pr) for forced convection Nu = f(Ra) for natural convection

Prandtl: Pr = μc_p/k = ν/α Ratio: momentum diffusivity / thermal diffusivity Pr << 1: liquid metals; Pr ≈ 1: gases; Pr >> 1: viscous oils

Biot: Bi = hL_c/k_solid Ratio: external convection / internal conduction Bi < 0.1: lumped capacitance valid; Bi >> 1: surface resistance negligible

Fourier: Fo = αt/L² Dimensionless time for transient conduction α = k/(ρc_p) = thermal diffusivity

Rayleigh: Ra = Gr×Pr = gβΔTL³/(να) Natural convection driver (Gr = Grashof number = gβΔTL³/ν²) Ra < 10⁹: laminar natural convection; Ra > 10⁹: turbulent

Stanton: St = Nu/(Re×Pr) = h/(ρVc_p) Heat transfer efficiency in forced convection

Mass Transfer

Schmidt: Sc = ν/D_AB = μ/(ρD_AB) Momentum / mass diffusivity (analogous to Pr) Sherwood: Sh = k_m L/D_AB (analogous to Nu) Lewis: Le = Sc/Pr = α/D_AB

Similarity Requirements

Types of Similarity

Geometric: all length ratios same (λ = L_model/L_prototype) Kinematic: velocity ratios at corresponding points same Dynamic: force ratios at corresponding points same (dimensionless groups match)

Complete Similarity

All relevant Pi groups must match between model and prototype. Often IMPOSSIBLE to match all groups simultaneously.

Reynolds similarity (viscous flow): Re_model = Re_prototype V_m L_m / ν_m = V_p L_p / ν_p V_m/V_p = (ν_m/ν_p)(L_p/L_m) = (ν_m/ν_p)/λ

Froude similarity (gravity waves, free surface): Fr_m = Fr_p V_m/√(gL_m) = V_p/√(gL_p) V_m/V_p = √(L_m/L_p) = √λ

Conflict: Can't match Re and Fr simultaneously in same fluid → compromise or choose which governs, neglect other Ship testing: Froude scaling (wave drag dominates); add Re correction

Wind Tunnel Testing

Match Re: ρ_m V_m L_m / μ_m = ρ_p V_p L_p / μ_p If same fluid (air), small model: V_m must be much higher (may exceed Ma limits) Solution: pressurized wind tunnel (higher ρ → lower V for same Re)

Mach scaling (compressible): must match Ma AND Re simultaneously Use same gas or choose model scale and accept Re mismatch at low Ma

Froude Model Scale Results

Velocity: V_m = V_p × λ^(1/2) Time: t_m = t_p × λ^(1/2) Force: F_m = F_p × λ³ (ρ same) Pressure: P_m = P_p × λ (ΔP scales with λ) Power: P_m = P_p × λ^(7/2)

Physical Meaning and Application

Self-Similarity

Blasius boundary layer: velocity profile u/U_∞ = f(η) where η = y√(U_∞/νx) One universal curve replaces infinite family of profiles

Scaling Laws

Heat exchanger: Nu = C × Re^m × Pr^n (Dittus-Boelter m=0.8, n=0.4) Allows prediction: scale correlation from small test to large system

Allometric scaling (biology/structures): Natural frequency: f ∝ 1/L (instruments, structures) Stiffness to weight: I ∝ L⁴, mass ∝ L³ → deflection ∝ L (geometrically similar beams)

Output

Provide: dimensionless Pi groups (n-r count), dominant governing group identification, model-to-prototype velocity/force/time scaling factors, similarity type (Re/Fr/Ma), prediction of prototype behavior from model measurements.