Which dimensionless number measures the ratio of inertial to gravitational forces?

Which dimensionless number measures the ratio of inertial to gravitational forces?

• A. Reynolds number
• B. Mach number
• C. Froude number
• D. Prandtl number

Answer: (C)

Understanding dimensionless numbers in fluid mechanics often comes down to identifying which forces you’re comparing. The Froude number does exactly that: it compares inertial forces to gravitational forces. It is defined as Fr = U / sqrt(gL), where U is a characteristic velocity, g is gravitational acceleration, and L is a characteristic length. Physically, a small Froude number means gravity dominates over inertia—the flow behaves as if energy losses and surface curvature are strongly resisted by gravity, leading to subcritical, more gently varying free surfaces. A large Froude number means inertia dominates and the surface responds more dynamically, producing phenomena like rapid surfaces changes or supercritical flow. This concept is especially important in open-channel flow, ship hydrodynamics, and wave-generation problems, where scaling and similarity rely on matching Fr between model and reality. In contrast, other common dimensionless numbers compare different balances: the Reynolds number compares inertial to viscous effects, the Mach number compares inertial to compressibility effects, and the Prandtl number compares momentum diffusivity to thermal diffusivity. Since the question asks for the ratio of inertial to gravitational forces, the Froude number is the appropriate measure.

Understanding dimensionless numbers in fluid mechanics often comes down to identifying which forces you’re comparing. The Froude number does exactly that: it compares inertial forces to gravitational forces. It is defined as Fr = U / sqrt(gL), where U is a characteristic velocity, g is gravitational acceleration, and L is a characteristic length.

Physically, a small Froude number means gravity dominates over inertia—the flow behaves as if energy losses and surface curvature are strongly resisted by gravity, leading to subcritical, more gently varying free surfaces. A large Froude number means inertia dominates and the surface responds more dynamically, producing phenomena like rapid surfaces changes or supercritical flow. This concept is especially important in open-channel flow, ship hydrodynamics, and wave-generation problems, where scaling and similarity rely on matching Fr between model and reality.

In contrast, other common dimensionless numbers compare different balances: the Reynolds number compares inertial to viscous effects, the Mach number compares inertial to compressibility effects, and the Prandtl number compares momentum diffusivity to thermal diffusivity. Since the question asks for the ratio of inertial to gravitational forces, the Froude number is the appropriate measure.