Dimensional analysis

Why it matters

Most fluid-flow and heat-transfer problems in industry are too complex to solve exactly, so engineers rely on experiments and correlations. Dimensional analysis tells you which combinations of variables actually matter, cuts the number of experiments drastically, and lets a small model in a laboratory predict the behaviour of a full-size pump, ship, spillway or heat exchanger.

Key ideas

Dimensions and units. Every physical quantity can be expressed in terms of a few fundamental dimensions: mass M, length L, time T and temperature Θ (in mechanics M, L, T are enough). Examples: velocity [L T⁻¹], force [M L T⁻²], pressure [M L⁻¹ T⁻²], dynamic viscosity [M L⁻¹ T⁻¹], kinematic viscosity [L² T⁻¹], surface tension [M T⁻²], power [M L² T⁻³].

Dimensional homogeneity. A physically meaningful equation must have the same dimensions in every additive term. This is a quick check on any formula you derive or remember: if the two sides do not match, the formula is wrong. Empirical formulas with dimensional constants (for example, some open-channel formulas) are valid only in the units for which they were written.

Rayleigh's method. Assume the dependent variable is a product of powers of the independent variables, write the dimensional equation, and equate exponents of M, L and T. It works well when there are at most about four variables.

Buckingham π theorem. If a problem involves n variables that are described by m fundamental dimensions, it can be reduced to a relation between (n − m) independent dimensionless groups, called π terms. Procedure:

1. List all n relevant variables (missing one gives a wrong result; including an irrelevant one only adds a spare group).
2. Choose m repeating variables that together contain all the fundamental dimensions but do not form a dimensionless group among themselves. A usual choice: one geometric variable (D or L), one kinematic (V), one fluid property (ρ). Do not choose the dependent variable as a repeating variable.
3. Form each π term as the product of the repeating variables (with unknown exponents) and one remaining variable; solve the exponents so that the group is dimensionless.
4. Write the result as π₁ = φ(π₂, π₃, …). The function φ must come from experiment.

Important dimensionless numbers (ratio of forces or effects).

• Reynolds number Re = ρVL/μ: inertia / viscous force — pipe flow, boundary layers, flow around bodies.
• Froude number Fr = V/√(gL): inertia / gravity — free-surface flows, ships, spillways.
• Euler number Eu = Δp/(ρV²): pressure / inertia force — pressure drops, pumps.
• Weber number We = ρV²L/σ: inertia / surface tension — sprays, droplets, coating.
• Mach number Ma = V/c: inertia / compressibility — high-speed gas flow.
• In heat transfer: Nusselt Nu = hL/k (convective / conductive heat transfer at the surface), Prandtl Pr = ν/α (momentum diffusivity / thermal diffusivity), Biot Bi = hL/k_solid (internal conductive resistance / external convective resistance), Grashof Gr (buoyancy / viscous force).

Similitude and model testing. A model predicts the prototype only if they are:

• geometrically similar (all lengths in the same ratio L_r),
• kinematically similar (velocity ratios the same at corresponding points),
• dynamically similar (force ratios, i.e. the governing dimensionless numbers, the same). In practice only the dominant force ratio can be matched. Fully enclosed flows (pipes, valves, submarines, aircraft at low speed) use Reynolds similarity. Free-surface flows (ships' wave resistance, spillways, weirs) use Froude similarity. Matching Re and Fr together with the same fluid is impossible, which is why ship resistance is split into a friction part and a wave part.

Froude-model scale ratios (with L_r = L_p/L_m): velocity ratio = √L_r, time ratio = √L_r, discharge ratio = L_r^2.5, force ratio = L_r³ (same fluid).

Formulas

number of π terms = n − m

• n: number of variables; m: number of fundamental dimensions.

Re = ρ·V·L / μ = V·L / ν; Fr = V / √(g·L); Eu = Δp / (ρ·V²); We = ρ·V²·L / σ; Ma = V / c

• ρ (kg/m³), V (m/s), L characteristic length (m), μ (Pa·s), ν (m²/s), Δp (Pa), σ (N/m), c speed of sound (m/s).

(V·L/ν)_model = (V·L/ν)_prototype

• Reynolds similarity, enclosed flows.

(V/√(gL))_model = (V/√(gL))_prototype, giving V_r = √L_r, Q_r = L_r^2.5

• Froude similarity, free-surface flows, same g.

F = ρ·V²·D²·φ(ρ·V·D/μ)

• General form of drag on a body (from the π theorem).

Worked examples

Example 1 (standard): drag on a sphere by the π theorem. Given: the drag F on a smooth sphere depends on its diameter D, the velocity V, and the 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 and there are 5 − 3 = 2 π terms.
2. Repeating variables: D [L], V [L T⁻¹], ρ [M L⁻³].
3. π₁ = D^a·V^b·ρ^c·F. Dimensions: L^a (L T⁻¹)^b (M L⁻³)^c (M L T⁻²) = M⁰L⁰T⁰. M: c + 1 = 0 → c = −1. T: −b − 2 = 0 → b = −2. L: a + b − 3c + 1 = 0 → a = −2. So π₁ = F / (ρ·V²·D²).
4. π₂ = D^a·V^b·ρ^c·μ with μ = [M L⁻¹ T⁻¹]. M: c + 1 = 0 → c = −1. T: −b − 1 = 0 → b = −1. L: a + b − 3c − 1 = 0 → a = −1. So π₂ = μ/(ρVD), the inverse of Re. Answer: F/(ρV²D²) = φ(Re), i.e. F = ρV²D²·φ(ρVD/μ) — a drag coefficient that depends only on Reynolds number.

Example 2 (GATE level): model of an oil pipeline. Given: an oil pipeline (D = 0.4 m, V = 2 m/s, ν = 1 × 10⁻⁵ m²/s, ρ = 900 kg/m³) is studied with a 1:4 scale model using water (ν = 1 × 10⁻⁶ m²/s, ρ = 1000 kg/m³). The model shows a pressure drop of 2 kPa over its scaled length. Find the model velocity and the prototype pressure drop.

1. Enclosed flow → Reynolds similarity. Re_p = V_p·D_p/ν_p = 2 × 0.4/(1 × 10⁻⁵) = 80 000.
2. D_m = 0.4/4 = 0.1 m; V_m = Re_p·ν_m/D_m = 80 000 × 1 × 10⁻⁶/0.1 = 0.8 m/s.
3. Same Euler number: Δp_p/(ρ_p·V_p²) = Δp_m/(ρ_m·V_m²).
4. Δp_p = 2000 × (900 × 2²)/(1000 × 0.8²) = 2000 × 3600/640 = 11 250 Pa Answer: V_m = 0.8 m/s; Δp_p ≈ 11.25 kPa

Example 3 (quick): Froude model of a spillway. At 1:25 scale the discharge ratio is Q_r = 25^2.5 = 3125 and the velocity ratio is √25 = 5. A model discharge of 0.5 L/s therefore represents 3125 × 0.0005 = 1.5625 m³/s, i.e. about 1.56 m³/s on the prototype.

Common mistakes

• Choosing the dependent variable (for example, the force or pressure drop) as a repeating variable.
• Choosing repeating variables that by themselves form a dimensionless group, or that do not contain all of M, L and T.
• Counting dimensions wrongly: in heat-transfer problems temperature Θ may be a fourth dimension.
• Using Reynolds similarity for a free-surface flow (or Froude for a pipe).
• Inverting the scale ratio: L_r = L_p/L_m is greater than 1 for a smaller model.
• Forgetting that φ is unknown; dimensional analysis gives the form of the relation, not the constants.

For GATE PI

Questions ask for the number of π terms, the dimensions of a quantity (viscosity, surface tension, power), the physical meaning of Re, Fr, Nu, Pr and Bi, and model-to-prototype conversions under Reynolds or Froude similarity (velocity, discharge, force, time ratios). Practise deriving one π term by equating exponents in under a minute and memorise the scale ratios for Froude models.

Quick check

1. What are the dimensions of dynamic viscosity?
2. A problem has 7 variables in M, L, T. How many π terms?
3. Which similarity law governs a model of a ship's wave resistance?
4. Under Froude similarity at 1:16 scale, what is the velocity ratio V_p/V_m?
5. What does the Prandtl number compare?

Answers: 1. M L⁻¹ T⁻¹ 2. 4 3. Froude 4. 4 5. Momentum diffusivity to thermal diffusivity (ν/α)