Verify that pressure head $p/(\rho g)$, velocity head $V^2/(2g)$, and elevation $z$ all have dimensions of length.

A proposed relation states $Q=AV^2$. Determine whether it can represent volumetric discharge.

Drag force $F$ on a body depends on fluid density $\rho$, velocity $V$, characteristic length $L$, and dynamic viscosity $\mu$. Derive a useful pair of dimensionless groups.

Suppose pressure drop depends on $\Delta p$, density $\rho$, velocity $V$, pipe diameter $D$, length $L$, dynamic viscosity $\mu$, and roughness height $\varepsilon$. Determine the number of independent Pi groups and identify a physically useful set.

A spillway model is built at model-to-prototype geometric scale $1:10$. Model discharge is $0.500\text{ m}^3/\text{s}$. Under Froude similarity with equal gravity, determine prototype discharge using $\lambda=L_p/L_m=10$.

A $1:25$ free-surface model has measured velocity $V_m=1.20\text{ m/s}$ and wave travel time $t_m=10.0\text{ s}$. Determine corresponding prototype values.

A $1:20$ Froude model uses the same fluid as the prototype. A model pressure difference is $2.00\text{ kPa}$, a measured force is $12.0\text{ N}$, and model power is $20.0\text{ W}$. Determine corresponding prototype values.

A $1:5$ pipe-flow model uses the same fluid as the prototype. Model velocity is $10.0\text{ m/s}$ and model discharge is $0.0200\text{ m}^3/\text{s}$. Determine prototype velocity and discharge under Reynolds similarity.

A $1:10$ air model is proposed for a water prototype. Take $\nu_m=1.50\times10^{-5}\text{ m}^2/\text{s}$ for air, $\nu_p=1.00\times10^{-6}\text{ m}^2/\text{s}$ for water, and prototype velocity $V_p=2.00\text{ m/s}$. Find the model velocity required for Reynolds similarity.

A $1:20$ free-surface model uses the same liquid and is operated under exact Froude similarity. The prototype Reynolds number is $Re_p=2.00\times10^6$. Determine the model Reynolds number.

Water flows at $V=2.00\text{ m/s}$ with characteristic length $L=0.100\text{ m}$. Use $\rho=1000\text{ kg/m}^3$, $\mu=0.00100\text{ Pa}\cdot\text{s}$, $g=9.81\text{ m/s}^2$, and $\sigma=0.0720\text{ N/m}$. Determine $Re$, $Fr$, and $We$.

Using $\lambda=L_p/L_m$, show why a reduced model using the same fluid and gravity cannot generally satisfy both exact Froude and exact Reynolds similarity.

A river model uses prototype-to-model horizontal scale $\lambda_H=100$ and vertical scale $\lambda_V=20$. The prototype bed slope is $S_p=0.00100$. Determine the model bed slope.

A $1:30$ Froude model uses a liquid of density $\rho_m=1000\text{ kg/m}^3$ to represent a prototype fluid of density $\rho_p=850\text{ kg/m}^3$. A model pressure difference is $1.20\text{ kPa}$ and model force is $50.0\text{ N}$. Determine corresponding prototype pressure difference and force.