Dimensional analysis and similitude

Why it matters

Many flow problems — drag on a vehicle, losses in a valve, performance of a new pump — cannot be solved exactly, so they are measured. Dimensional analysis tells you which few dimensionless groups govern the result, cutting the number of experiments drastically, and similitude tells you how to run a small model so that its measurements scale correctly to the full-size prototype.

Key ideas

Dimensions and homogeneity. Every physical quantity can be expressed in fundamental dimensions: mass M, length L, time T (and temperature θ for heat transfer). A correct physical equation is dimensionally homogeneous — every term has the same dimensions. Checking this catches algebra errors and tells you the units of unknown constants. Examples: force [MLT⁻²], pressure [ML⁻¹T⁻²], dynamic viscosity [ML⁻¹T⁻¹], kinematic viscosity [L²T⁻¹], surface tension [MT⁻²], power [ML²T⁻³].

Rayleigh's method. Write the quantity as a product of powers of the variables, equate exponents of M, L and T, and solve. It works well when there are few variables (up to about four).

Buckingham π theorem. If a physical relation involves n variables and the dimensional matrix has rank m (usually the number of fundamental dimensions, 3), the relation can be rewritten in terms of (n − m) independent dimensionless π groups. Procedure:

1. List all relevant variables, including the dependent one.
2. Choose m repeating variables that together contain all the fundamental dimensions, are not themselves able to form a dimensionless group, and preferably are a geometric, a kinematic and a fluid property (for example D, V, ρ). Do not choose the dependent variable as a repeating variable.
3. Combine the repeating variables with each remaining variable in turn to make it dimensionless.
4. The groups can be inverted, multiplied or combined to give familiar numbers. The theorem tells you which groups matter, not the form of the function; experiments supply that.

Important dimensionless numbers (each is a force ratio):

• Reynolds number Re = ρVL/μ: inertia/viscous. Governs pipe flow, boundary layers, submerged bodies.
• Froude number Fr = V/√(gL): inertia/gravity. Governs free-surface flows — ships, spillways, open channels.
• Euler number Eu = Δp/(ρV²): pressure/inertia. The usual dependent group for pressure drop.
• Weber number We = ρV²L/σ: inertia/surface tension. Droplets, sprays, thin films.
• Mach number Ma = V/c: inertia/compressibility. Significant above about 0.3 in gases.

Similitude. A model represents a prototype when there is:

• Geometric similarity: all lengths scale by the same ratio L_r = L_m/L_p (including roughness, ideally).
• Kinematic similarity: velocity ratios are the same at corresponding points, so streamline patterns match.
• Dynamic similarity: ratios of all forces are the same, which means the relevant dimensionless numbers are equal in model and prototype. Geometric similarity is a prerequisite; with it, dynamic similarity gives kinematic similarity.

Model laws in practice. It is usually impossible to match Re and Fr together with the same fluid, so the dominant force decides:

• Reynolds model law for fully enclosed flows (pipes, submarines, aircraft at low Mach): V_m = V_p·(L_p/L_m)·(ν_m/ν_p).
• Froude model law for free-surface flows (ships, spillways, dams): V_r = √L_r; then time scales as √L_r, discharge as L_r^2.5, force as L_r³ and power as L_r^3.5 (same fluid).
• Mach law for high-speed compressible flows. Ship-model testing separates resistance into a friction part (estimated from flat-plate data at each Reynolds number) and a wave part (scaled by Froude law). Distorted models (different horizontal and vertical scales) are used for rivers and harbours.

Formulas

Number of π groups = n − m

• n = number of variables, m = rank of the dimensional matrix (usually 3).

Re = ρ·V·L / μ = V·L / ν

• ρ (kg/m³), V (m/s), L = characteristic length (m), μ (Pa·s), ν (m²/s).

Fr = V / √(g·L), Eu = Δp / (ρ·V²), We = ρ·V²·L / σ, Ma = V / c

• Δp (Pa), σ (N/m), c = speed of sound (m/s). All dimensionless.

V_m / V_p = (L_p / L_m) · (ν_m / ν_p) (Reynolds law)

V_r = √L_r, Q_r = L_r^2.5, F_r = L_r³, P_r = L_r^3.5 (Froude law, same fluid, g the same)

• Subscript r = model/prototype ratio.

F = ρ·V²·D²·φ(Re) (drag on a body, from π analysis)

Worked examples

Example 1 — drag on a sphere by the π theorem (standard). The drag F on a smooth sphere depends on its diameter D, speed V, fluid density ρ and viscosity μ. Find the dimensionless form.

1. Variables: F, D, V, ρ, μ, so n = 5; dimensions M, L, T, so m = 3; number of groups = 2.
2. Repeating variables: D, V, ρ.
3. π₁ = F·D^a·V^b·ρ^c. Dimensions: MLT⁻² · L^a · (LT⁻¹)^b · (ML⁻³)^c = M⁰L⁰T⁰.
   • M: 1 + c = 0, so c = −1. T: −2 − b = 0, so b = −2. L: 1 + a + b − 3c = 0, so a = −2.
   • π₁ = F/(ρ·V²·D²).
4. π₂ = μ·D^a·V^b·ρ^c: M: 1 + c = 0; T: −1 − b = 0; L: −1 + a + b − 3c = 0, so c = −1, b = −1, a = −1. π₂ = μ/(ρVD) = 1/Re. Answer: F/(ρV²D²) = φ(Re), i.e. F = ρV²D²·φ(Re). This is why drag coefficients are plotted against Reynolds number.

Example 2 — Froude and Reynolds models (GATE level). (a) A 1:25 ship model is tested in fresh water for a prototype speed of 10 m/s. The measured wave resistance is 50 N. Find the model speed and the prototype wave resistance (neglect the difference between sea and fresh water density).

1. Froude law: V_m = V_p·√L_r = 10 × √(1/25) = 2.0 m/s.
2. Force scale: F_r = L_r³, so F_p = 50 × 25³ = 781 250 N. Answer: V_m = 2 m/s; wave resistance ≈ 781 kN.

(b) Oil (ν = 1 × 10⁻⁵ m²/s, ρ = 900 kg/m³) flows at 2 m/s in a 0.5 m pipe. A 0.1 m model uses water (ν = 1 × 10⁻⁶ m²/s, ρ = 1000 kg/m³). Find the model speed for dynamic similarity, and the prototype pressure drop if the model shows 12 kPa between corresponding sections.

1. Reynolds law: V_m = V_p·(D_p/D_m)·(ν_m/ν_p) = 2 × 5 × 0.1 = 1.0 m/s.
2. Equal Euler numbers: Δp_p = Δp_m × (ρ_p·V_p²)/(ρ_m·V_m²) = 12 × (900 × 4)/(1000 × 1) = 43.2 kPa. Answer: V_m = 1.0 m/s; Δp_p = 43.2 kPa.

Common mistakes

• Choosing the dependent variable (the one you want to find) as a repeating variable.
• Choosing repeating variables that together lack one of M, L or T, or that already form a dimensionless group (such as V, L and t).
• Applying Froude law to a closed pipe or Reynolds law to a ship's wave resistance.
• Forgetting that in Froude scaling time scales with √L_r, not L_r, when converting model test durations.
• Using the scale ratio upside down: always write ratio = model/prototype and keep it consistent.
• Assuming a dimensionless group fixes the function; the π theorem only gives the grouping.

For GATE ME

Expect questions on the number of π terms, the dimensions of viscosity, surface tension or power, identifying the right dimensionless number for a situation, and model-law numericals: model speed, discharge, force or power scaling under Froude or Reynolds similarity. Practise the exponent algebra for π groups quickly and memorise the Froude scaling exponents.

Quick check

1. How many π groups describe a relation among 7 variables with 3 fundamental dimensions?
2. Which dimensionless number governs flow over a spillway?
3. What are the dimensions of dynamic viscosity?
4. In a 1:16 Froude model, what is the velocity ratio?
5. Which force ratio is the Weber number?

Answers: 1. 4; 2. Froude number; 3. ML⁻¹T⁻¹; 4. V_m/V_p = 1/4; 5. inertia force to surface-tension force.