Fluid dynamics — where do both Reynolds and Froude numbers matter? In which of the following real-world flows are both viscous/inertial effects (Reynolds number) and gravity/free-surface wave effects (Froude number) simultaneously significant for modelling and performance assessment?

Introduction / Context:
Similitude in fluid mechanics relies on non-dimensional numbers. Reynolds number (Re) captures the ratio of inertial to viscous forces and governs boundary-layer behaviour and frictional resistance. Froude number (Fr) captures the ratio of inertia to gravity and governs free-surface waves and hydrostatic effects. Some applications are governed mainly by one; others require attention to both for realistic prediction and scaling.

Given Data / Assumptions:

• Ship motion at the free surface generates waves (gravity-controlled) and experiences viscous drag (boundary-layer skin friction and eddy resistance).
• Submerged bodies far from the free surface do not create surface waves, so Fr is not the key parameter.
• High-speed missiles in air are dominated by Mach/compressibility, not Froude.
• Spillway free-surface flows are typically treated as Froude-dominated; Re is very high and not matched in small models.

Concept / Approach:
For free-surface ships, wave-making resistance depends on Fr similarity, while viscous resistance depends on Re. In model tests, Fr similarity is prioritised to capture wave systems; Re cannot be simultaneously matched at practical scale, so viscous effects are corrected empirically (e.g., ITTC friction lines). Thus both numbers matter in design/performance assessment, even if only one is matched exactly in a towing tank.

Step-by-Step Solution:
Identify if a free surface is dynamically important → if yes, Froude matters.Check whether viscous boundary layers and form drag contribute materially → if yes, Reynolds matters.Evaluate options: only a surface ship clearly involves both wave-making (Fr) and viscous resistance (Re) as first-order effects.

Verification / Alternative check:
Standard ship hydrodynamics separates resistance into frictional (Re-sensitive) and residuary (wave-making, Fr-sensitive) components; towing-tank practice confirms dual relevance.

Why Other Options Are Wrong:
Submarine at depth: negligible surface waves → Froude not governing.
Missile in air: Mach number and lift/drag polars dominate; Froude is not the scaling parameter.
Spillway: Froude similarity is primary; high Re makes viscous similarity secondary and usually not matched.
Full pipe flow: no free surface → Froude irrelevant.

Common Pitfalls:

• Assuming Re must be perfectly matched in ship models; in practice, viscous corrections are applied.
• Using Fr for closed-conduit internal flows where no free surface exists.

Final Answer:
Motion of a surface ship in deep seas (wave-making and viscous resistance)