Problem 1: Dimensional Homogeneity of Bernoulli Terms

Verify that $p/\rho g$, $V^2/2g$, and $z$ all have dimensions of length.

Problem 2: Buckingham Pi Groups for Drag

Drag force $F$ on a body depends on fluid density $\rho$, velocity $V$, characteristic length $L$, and dynamic viscosity $\mu$. Use Buckingham Pi reasoning to identify two dimensionless groups.

Problem 3: Froude-Scaled Spillway Discharge

A spillway model is built at model-to-prototype scale $1:10$. Model discharge is $0.50\text{ m}^3/\text{s}$. Determine prototype discharge under Froude similarity. Define $\lambda_L=L_p/L_m=10$.

Problem 4: Froude Velocity and Time Scales

A $1:25$ free-surface model has velocity $1.2\text{ m/s}$ and an observed wave travel time of $10\text{ s}$. Determine prototype velocity and time.

Problem 5: Reynolds Similarity with the Same Fluid

A $1:5$ pipe-flow model uses the same fluid as the prototype. Model velocity is $10\text{ m/s}$ and model discharge is $0.020\text{ m}^3/\text{s}$. Determine prototype velocity and discharge under Reynolds similarity.

Problem 6: Reynolds Similarity with Different Fluids

A $1:10$ air model is proposed for a water prototype. Take $\nu_{air}=1.5\times10^{-5}\text{ m}^2/\text{s}$, $\nu_{water}=1.0\times10^{-6}\text{ m}^2/\text{s}$, and prototype velocity $2.0\text{ m/s}$. Find the air-model velocity required for Reynolds similarity.

Problem 7: Froude Force and Power Scales

A geometrically similar hydraulic model has scale $1:20$ and uses the same fluid as the prototype. A measured model force is $12\text{ N}$ and model power is $20\text{ W}$. Determine corresponding prototype force and power under Froude similarity.

Problem 8: Reynolds, Froude, and Weber Numbers

Water flows at $2.0\text{ m/s}$ with characteristic length $0.10\text{ m}$. Use $\rho=1000\text{ kg/m}^3$, $\mu=0.001\text{ Pa}\cdot\text{s}$, $g=9.81\text{ m/s}^2$, and surface tension $\sigma=0.072\text{ N/m}$. Determine $Re$, $Fr$, and $We$.