Dimensional Analysis and Similitude Questions and Answers

Answer: B

The Reynolds number is the dimensionless number used to predict the transition between laminar and turbulent flow. It is the ratio of the inertia forces to the viscous forces, \(Re = \frac{\rho V D}{\mu}\), and for flow in pipes the flow becomes turbulent above a critical value of about 4000. The Prandtl number compares the diffusion of momentum and heat, the Froude number compares inertia and gravity forces, and the Nusselt number compares convective and conductive heat transfer, so none of them predicts the transition.

Answer: B

The ratio of inertial to viscous forces is the Reynolds number, \(Re = \rho VL/\mu\).

It decides whether a flow is laminar (low \(Re\), viscous forces dominate) or turbulent (high \(Re\), inertia dominates), as in pipe flow, where the change happens at about \(Re = 2300\).

The Froude number compares inertial forces with gravity forces and governs open-channel and free-surface flow. The power number relates the power drawn by an impeller or agitator to the fluid density, rotational speed and impeller diameter. The Biot number is a heat-transfer group comparing conduction resistance inside a body with convection resistance at its surface. None of these involves viscosity in the way the question describes.

Answer: C

The prototype velocity is 10 m \(\text{s}^{-1}\).

Froude similarity requires equal \(V/\sqrt{gL}\) in model and prototype. With the same \(g\):

\[ \begin{aligned} V_{p} &= V_{m}\sqrt{L_{p}/L_{m}} \\ &= 1\times\sqrt{100} = 10\ \text{m/s} \end{aligned} \]

100 scales velocity with length directly, as Reynolds or geometric reasoning might suggest, and 0.1 inverts the ratio. 1 would hold only for a full-scale model.

Answer: D

\(V/\sqrt{Lg}\) is the Froude number.

The Froude number compares inertia forces with gravity forces, and governs free-surface flows such as open channels, ship waves and spillways. Its square is \(V^{2}/(Lg)\), the ratio of inertial to gravitational effects.

The Weber number \(\rho V^{2}L/\sigma\) compares inertia with surface tension. The Euler number \(\Delta p/(\rho V^{2})\) compares pressure forces with inertia, and the cavitation number \((p - p_{v})/(\tfrac{1}{2}\rho V^{2})\) is a form of it built on the vapour pressure. None of them involves g.

Answer: C

The drag on the prototype is about 1.08 N.

Dynamic similarity for a submerged body means equal Reynolds numbers. That fixes the model speed:

\[ \begin{aligned} \frac{1.2(1)(4)}{1.8 \times 10^{-5}} &= \frac{1000\,V_{m}(0.1)}{10^{-3}} \\ 2.667 \times 10^{5} &= 10^{5}V_{m} \Rightarrow V_{m} = 2.667\ \text{m/s} \end{aligned} \]

With equal drag coefficients, drag scales as \(\rho V^{2}D^{2}\):

\[ \begin{aligned} \frac{F_{p}}{F_{m}} &= \frac{1.2(1)^{2}(4)^{2}}{1000(2.667)^{2}(0.1)^{2}} = \frac{19.2}{71.1} \\ F_{p} &= 4 \times 0.27 \approx 1.08\ \text{N} \end{aligned} \]

4 N assumes the drag is unchanged between model and prototype, and the other values come from scaling with the wrong variables.

Answer: C

The Froude number matters most in free surface flows.

The Froude number, Fr = V/√(gL), compares inertia with gravity. Gravity governs flows with a free surface, such as open channels, ship waves and hydraulic jumps, where Fr decides whether the flow is sub- or supercritical.

Capillary flows are governed by surface tension (Weber and capillary numbers), creeping flows by viscosity (very small Reynolds number), and compressible flows by the Mach number. Gravity plays little part in any of these.

Answer: A

The correct combination is 1-iii, 2-i, 3-ii, 4-iv.

• Reynolds number (inertia/viscous forces) governs viscous drag.
• Froude number (inertia/gravity) governs free-surface flows, and so the wave drag of ships.
• Mach number (flow speed/speed of sound) measures compressibility of the flow.
• Weber number (inertia/surface tension) governs droplet break-up and so spray formation.

The other options misplace at least two of these pairs; for example, one assigns the Reynolds number to wave drag and the Froude number to compressible flow, and another pairs Mach number with both spray formation and viscous drag.

Answer: C

The new speed is \(c' = \frac{c}{\sqrt{2}}\).

By dimensional analysis, the only combination of \(\rho\) (kg/m³), \(\lambda\) (m) and \(\sigma\) (N/m = kg/s²) with units of speed is

\[ c \propto \sqrt{\frac{\sigma}{\rho\lambda}} \]

so \(c\) varies as \(\sqrt{\sigma}\). Halving \(\sigma\) with \(\rho\) and \(\lambda\) fixed multiplies c by \(1/\sqrt{2}\). \(\sqrt{2}c\) and \(2c\) would mean the speed rises when the restoring force weakens, and \(c' = c\) ignores the dependence on surface tension altogether.

Answer: B

The number is the Strouhal number.

\[ St = \frac{fL}{V} \]

It compares the frequency \(f\) of an unsteady flow with the flow's own time scale \(V/L\), and is the standard measure of periodic phenomena such as vortex shedding behind a cylinder, where \(St \approx 0.2\) over a wide range of Reynolds numbers.

The Froude number compares inertia with gravity (free-surface flows and waves). The Peclet number compares advection with diffusion of heat or mass. The Lewis number compares thermal diffusivity with mass diffusivity. None of them contains a frequency.

Answer: D

The tunnel speed should be 128 m/s.

For flow around a car the governing similarity parameter is the Reynolds number. With the same fluid, air, in the tunnel, \(\nu\) is unchanged, so

\[ \begin{aligned} V_m L_m &= V_p L_p \\ V_m &= 16 \times \frac{L_p}{L_m} = 16 \times 8 = 128\ \text{m/s} \end{aligned} \]

16 m/s would keep the speed but not the Reynolds number, and 2 m/s scales the speed down instead of up. 64 m/s would match Reynolds number only for a quarter-scale model. (At 128 m/s compressibility starts to matter, which is why real tunnel tests often relax full Reynolds matching.)