Which pairing of dimensionless number and force ratio is correct?

Mach: inertia / surface tension

For which flow is the Froude number the key similarity parameter?

Spillway or ship with a free surface

What does dynamic similarity require and imply?

Equal relevant dimensionless numbers; it implies kinematic and geometric similarity

Only equal shapes; it implies equal forces

Equal velocities only; it does not imply geometric similarity

Why can Re and Fr not both be matched in a model using the same fluid?

Fr matching needs V_r = sqrt(L_r) while Re matching needs V_r = 1/L_r, which agree only if L_r = 1

Re cannot be defined for a model

Gravity is different in the model

Practice · 8 marks

Dimensionless numbers and types of similarity

Short answer

Reynolds, Froude, Euler, Weber, and Mach numbers compare inertia to viscous, gravity, pressure, surface tension, and compressibility forces. Similarity requires geometric, kinematic, and dynamic matching. Complete similarity is often impossible because matching both Re and Fr requires conflicting velocity scales.

Define Reynolds, Froude, Euler, Weber and Mach numbers as ratios of forces and state where each is important. Explain geometric, kinematic and dynamic similarity, and explain why complete similarity is often impossible in model studies (incomplete similarity).

Dimensionless numbers

Each number compares the inertia force with one other force. (The Euler number is usually written the other way round, as pressure force to inertia force.)

Number	Expression	Force ratio	Important in
Reynolds	$Re = \dfrac{\rho V L}{\mu}$	Inertia / viscous	Pipe flow, boundary layer, flow around bodies
Froude	$Fr = \dfrac{V}{\sqrt{gL}}$	Inertia / gravity	Free surface flow: spillways, ships, channels
Euler	$Eu = \dfrac{\Delta p}{\rho V^2}$	Pressure / inertia	Cavitation, pressure-driven flow
Weber	$We = \dfrac{\rho V^2 L}{\sigma}$	Inertia / surface tension	Droplets, capillary waves, thin films
Mach	$Ma = \dfrac{V}{a}$	Inertia / elastic (compressibility)	High-speed gas flow

Types of similarity

Geometric similarity: the model and prototype have the same shape; all linear dimensions have the same scale ratio $L_r = L_p/L_m$ , and angles are equal.
Kinematic similarity: velocities at corresponding points are in the same direction and have the same ratio, so the streamline patterns are similar. This needs geometric similarity and a fixed time scale.
Dynamic similarity: forces at corresponding points have the same ratio (and the same direction). It requires the relevant dimensionless numbers of the model and prototype to be equal. Dynamic similarity implies kinematic and geometric similarity.

Incomplete similarity

In practice more than one force may matter, e.g. both gravity and viscosity for a ship. Keeping both $Re$ and $Fr$ equal would need a model fluid of unrealistic viscosity: for $Fr_m = Fr_p$ , $V_r = \sqrt{L_r}$ , while for $Re_m = Re_p$ with the same fluid, $V_r = 1/L_r$ . Both cannot be satisfied with the same fluid unless $L_r = 1$ .

Therefore only the dominant force is matched:

Gravity dominant (spillway, ship wave resistance, open channel): equate $Fr$ .
Viscosity dominant (pipe flow, submarine, aircraft at low speed): equate $Re$ .

The other effects are estimated by correction factors or by running tests at several scales. Such a model is said to have incomplete (partial) similarity.

✓ Answer checked by Claude Opus 5.5 (high effort)

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Dimensionless numbers and types of similarity

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Types of similarity

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