| 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
- List all n variables: variable list with units
- Identify r = number of primary dimensions involved (typically r ≤ 5)
- Choose r repeating variables (must span all dimensions; avoid choosing the dependent variable)
- Form (n-r) Pi groups by combining repeating variables with each remaining variable
- 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.
