Dimensional Analysis & Model Similitude Notes

Last reviewed 16 Sept 2026 · 7 min read

Dimensions

Physical quantities are expressed in the fundamental dimensions mass [M], length [L] and time [T] (temperature [θ] where needed).

Quantity	Dimensions	Quantity	Dimensions
Velocity	LT⁻¹	Force	MLT⁻²
Acceleration	LT⁻²	Pressure, stress	ML⁻¹T⁻²
Discharge	L³T⁻¹	Work, energy	ML²T⁻²
Kinematic viscosity	L²T⁻¹	Power	ML²T⁻³
Dynamic viscosity	ML⁻¹T⁻¹	Surface tension	MT⁻²
Density	ML⁻³	Specific weight	ML⁻²T⁻²
Angular velocity	T⁻¹	Bulk modulus	ML⁻¹T⁻²
Torque, moment	ML²T⁻²	Momentum	MLT⁻¹

Dimensional homogeneity (Fourier's principle): every term in a correct physical equation has the same dimensions. It lets you check equations, convert units and derive relations.

Rayleigh's method

Used when the dependent variable is a function of few (up to about 3–4) independent variables.

Write $X = k X_{1}^{a} X_{2}^{b} X_{3}^{c} \dots$
Substitute dimensions and equate powers of M, L and T.
Solve for the exponents.

Buckingham π-theorem

If a physical phenomenon involves $n$ variables containing $m$ fundamental dimensions, it can be expressed as a relation among $(n - m)$ dimensionless π-terms:

f (π_{1}, π_{2}, \dots, π_{n - m}) = 0

Choosing repeating variables ( $m$ of them, usually 3):

They must not include the dependent variable.
Together they must contain all fundamental dimensions.
They must not form a dimensionless group among themselves, and no two should have identical dimensions.
Choose one geometric property (length $D$ , $L$ ), one flow property (velocity $V$ ) and one fluid property (density $ρ$ ) — the usual set is $D$ , $V$ , $ρ$ .

Dimensionless numbers

FormulaImportant dimensionless numbers

Number	Definition	Force ratio	Significant in
Reynolds $R e$	$\frac{ρ V L}{μ}$	Inertia / viscous	Pipe flow, submerged bodies, low-speed aircraft, submarines
Froude $F r$	$\frac{V}{g L}$	√(Inertia / gravity)	Free-surface flows — channels, spillways, weirs, ships (wave resistance), hydraulic jumps
Euler $E u$	$\frac{V}{p / ρ}$	√(Inertia / pressure)	Flows dominated by pressure — cavitation, pipe flow with pressure changes
Weber $W e$	$\frac{V}{σ / ( ρ L )}$	√(Inertia / surface tension)	Capillary waves, droplets, very small heads over weirs
Mach $M$	$\frac{V}{C} = \frac{V}{K / ρ}$	√(Inertia / elastic)	Compressible flow — high-speed aircraft, projectiles, water hammer

Some texts define Froude, Euler, Weber and Mach numbers as the squares of the ratios above; the physical meaning is unchanged.

Similitude

A model is a scaled replica used to predict prototype behaviour. For complete similarity:

Geometric similarity — all linear dimensions in a constant ratio $L_{r} = L_{p} / L_{m}$ (areas $L_{r}^{2}$ , volumes $L_{r}^{3}$ ).
Kinematic similarity — velocities and accelerations at corresponding points in constant ratios (same streamline pattern).
Dynamic similarity — ratios of all forces at corresponding points equal, i.e. the relevant dimensionless numbers are equal in model and prototype.

It is usually impossible to equate all dimensionless numbers simultaneously; the dominant force decides the model law.

Model laws and scale ratios

FormulaFroude model law (gravity dominant;

g

same)

$(F r)_{m} = (F r)_{p}$ → with length scale $L_{r}$ :

Quantity	Scale ratio
Velocity $V_{r}$	$L_{r}^{1/2}$
Time $T_{r}$	$L_{r}^{1/2}$
Acceleration	1
Discharge $Q_{r}$	$L_{r}^{5/2}$
Force $F_{r}$	$L_{r}^{3}$ (same fluid)
Pressure intensity $p_{r}$	$L_{r}$
Work / energy	$L_{r}^{4}$
Power $P_{r}$	$L_{r}^{7/2}$

FormulaReynolds model law (viscosity dominant)

$(R e)_{m} = (R e)_{p}$ → $V_{r} = \frac{ν _{r}}{L _{r}}$

With the same fluid in model and prototype ( $ν_{r}$ = 1): $V_{r} = 1/ L_{r}$ , $T_{r} = L_{r}^{2}$ , $Q_{r} = L_{r}$ , $F_{r} = 1$ . Model velocities become very high for small models — a practical drawback (often use a different fluid, or a wind tunnel for water problems).

Euler law — $V_{r} = p_{r} / ρ_{r}$ ; used where pressure forces dominate and turbulence is fully developed.
Weber law — $V_{r} = σ_{r} / (ρ_{r} L_{r})$ .
Mach law — $V_{r} = K_{r} / ρ_{r}$ (e.g. aerodynamic testing, water hammer models).

Ship models: wave resistance obeys Froude's law but skin friction obeys Reynolds' law; both cannot be satisfied with water as the model fluid, so the model is run to Froude's law and friction resistance is calculated separately (Froude's method).

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Dimensional Analysis & Model Similitude