Dimensional analysis and similitude

Why it matters

Car bodies, pumps, turbochargers and spillways are developed on scale models in wind tunnels, water channels and test rigs long before the full-size item exists. Dimensional analysis tells you which groups of variables to test, and similitude tells you how to run the model and scale its results up. It also explains why the Moody chart and pump characteristic curves can be drawn once for a whole family of sizes.

Key ideas

• Dimensions. Fluid-mechanics quantities can be written in terms of mass M, length L and time T (plus temperature θ for heat transfer). Examples: velocity LT⁻¹, force MLT⁻², pressure ML⁻¹T⁻², dynamic viscosity ML⁻¹T⁻¹, kinematic viscosity L²T⁻¹, power ML²T⁻³.
• Dimensional homogeneity. Every term in a valid physical equation has the same dimensions. This is used to check equations and find the units of constants.
• Rayleigh's method. Assume the dependent variable is a product of powers of the independent variables, then equate the exponents of M, L and T. It works well when there are few variables (up to about four).
• Buckingham π theorem. If a problem involves n variables containing m fundamental dimensions, it can be written as a relation among (n − m) independent dimensionless π groups.
  • Choose m repeating variables that together contain all m dimensions and do not form a dimensionless group among themselves. Usually pick one geometric variable (D or L), one flow variable (V or N) and one fluid property (ρ).
  • Do not choose the dependent variable as a repeating variable.
  • Each remaining variable combined with the repeating set forms one π group.
• Important dimensionless numbers (each a force ratio):
  • Reynolds number Re: inertia / viscous force. Governs pipe flow, boundary layers and drag at low speed.
  • Froude number Fr: inertia / gravity. Governs free-surface flows — ships, spillways, open channels.
  • Euler number Eu: pressure / inertia force. A dimensionless pressure drop or pressure coefficient.
  • Weber number We: inertia / surface-tension force. Governs sprays, droplets and fuel atomisation.
  • Mach number Ma: inertia / elastic (compressibility) force. Governs high-speed gas flow; compressibility matters above Ma ≈ 0.3.
• Similitude between model (m) and prototype (p):
  • Geometric similarity: all length ratios equal (scale ratio L_r = L_p/L_m), including roughness ideally.
  • Kinematic similarity: velocity ratios (and flow directions) equal at corresponding points.
  • Dynamic similarity: ratios of all forces equal at corresponding points — achieved by matching the governing π groups. With geometric similarity, dynamic similarity gives kinematic similarity.
• Model laws. Match the dominant force ratio: Reynolds model law for enclosed or fully submerged flows (pipes, submarines, low-speed aerodynamics); Froude model law for free-surface flows; Mach law for high-speed gas flow. Usually you cannot match Re and Fr at once with the same fluid, so the model is "distorted" and corrections are applied.
• Turbomachines follow the same idea: geometrically similar pumps share the same flow coefficient Q/(ND³), head coefficient gH/(N²D²) and power coefficient P/(ρN³D⁵). These give the affinity laws used in the pump and turbine topics.

Formulas

Re = ρ·V·L/μ = V·L/ν ρ density (kg/m³), V velocity (m/s), L characteristic length (m), μ dynamic viscosity (Pa·s), ν kinematic viscosity (m²/s).

Fr = V/√(g·L) Eu = Δp/(ρ·V²) We = ρ·V²·L/σ Ma = V/c g = 9.81 m/s², Δp pressure difference (Pa), σ surface tension (N/m), c speed of sound (m/s).

Number of π groups = n − m n variables, m fundamental dimensions.

V_m = V_p·(L_p/L_m)·(ν_m/ν_p) Reynolds model law (Re_m = Re_p).

V_r = √L_r, Q_r = L_r^2.5, T_r = √L_r, F_r = L_r³ Froude model law with the same fluid and g; subscript r means prototype/model ratio; T is time.

F_p/F_m = (ρ_p/ρ_m)·(V_p/V_m)²·(L_p/L_m)² Force scaling when the force coefficient F/(ρV²L²) is matched.

Q/(N·D³), g·H/(N²·D²), P/(ρ·N³·D⁵) = constant Similar turbomachines; N speed (rev/s or rpm used consistently), D diameter (m), H head (m), P power (W).

Worked examples

Example 1 (standard) — Buckingham π for pipe pressure drop. Given: Δp in a pipe depends on D, L, V, ρ, μ and roughness ε. Find the π groups.

1. Variables: Δp, D, L, V, ρ, μ, ε → n = 7. Dimensions M, L, T → m = 3. So 7 − 3 = 4 π groups.
2. Repeating variables: D (L), V (LT⁻¹), ρ (ML⁻³).
3. π₁ = Δp·D^a·V^b·ρ^c: M: 1 + c = 0 → c = −1; T: −2 − b = 0 → b = −2; L: −1 + a + b − 3c = 0 → a = 0. So π₁ = Δp/(ρV²).
4. π₂ with μ (ML⁻¹T⁻¹): c = −1, b = −1, a = −1 → μ/(ρVD), i.e. 1/Re.
5. π₃ = L/D and π₄ = ε/D (already lengths). Answer: Δp/(ρV²) = f(ρVD/μ, L/D, ε/D). Since Δp is proportional to L/D, this becomes the Darcy friction factor f(Re, ε/D) — the Moody chart.

Example 2 (GATE level) — car model in a water tunnel. Given: a car 1:5 scale model (L_p/L_m = 5) is tested in water to simulate the prototype at 25 m/s in air. ν_air = 1.5 × 10⁻⁵ m²/s, ν_water = 1.0 × 10⁻⁶ m²/s, ρ_air = 1.2 kg/m³, ρ_water = 1000 kg/m³. The model drag is 200 N. Find the model speed and the prototype drag.

1. Reynolds law: V_m = V_p·(L_p/L_m)·(ν_m/ν_p) = 25 × 5 × (1.0 × 10⁻⁶/1.5 × 10⁻⁵) = 8.333 m/s.
2. With Re matched, the drag coefficient is the same, so F_p/F_m = (ρ_p/ρ_m)·(V_p/V_m)²·(L_p/L_m)².
3. F_p = 200 × (1.2/1000) × (25/8.333)² × 5² = 200 × 0.0012 × 9 × 25 = 54 N. Answer: V_m ≈ 8.33 m/s; prototype drag ≈ 54 N. In air, the same model would need 125 m/s (Ma ≈ 0.36), which brings in compressibility — the reason water tunnels are used.

Example 3 (Froude law). A 1:25 spillway model carries 0.2 m³/s. Prototype discharge Q_p = 0.2 × 25^2.5 = 0.2 × 3125 = 625 m³/s, and prototype velocities are √25 = 5 times the model's.

Common mistakes

• Choosing the dependent variable, or a set that is itself dimensionless, as repeating variables.
• Inverting the scale ratio — for Reynolds similarity in the same fluid, a smaller model needs a higher speed.
• Using the Reynolds law for a free-surface model (use Froude) or Froude for a closed-pipe model.
• Forgetting that kinematic viscosity, not dynamic viscosity, appears when the model fluid differs.
• Writing Q_r = L_r² or L_r³ under Froude scaling; it is L_r^2.5.

For GATE ME

Typical questions: the number of π groups, identifying a dimensionless group or the dimensions of a quantity, the model velocity or discharge under the Reynolds or Froude law, force or power scaling, and the meaning of Re, Fr, We, Ma and Eu. Practise writing M-L-T dimensions quickly and checking the exponent algebra.

Quick check

1. How many π groups for 6 variables involving M, L and T?
2. What are the dimensions of dynamic viscosity?
3. Which dimensionless number governs ship-model testing?
4. Under Froude scaling, what is the velocity ratio for a 1:16 model?
5. Water (ρ = 1000 kg/m³, μ = 0.001 Pa·s) flows at 2 m/s over a 0.5 m plate. What is Re_L? Answers: 1. 3. 2. ML⁻¹T⁻¹. 3. Froude number. 4. V_p/V_m = √16 = 4. 5. 1000 × 2 × 0.5/0.001 = 1.0 × 10⁶.