10 / CHAPTER 09
Vorticity, circulation, and potential flow
Describe rotation in a fluid and identify useful idealizations.
ANCHOR EQUATION
Vorticity
In symbols: ω = ∇×u
Vorticity is twice the local angular-velocity vector for a fluid element.
Dimensionless similarity transfers experiments
A model nozzle can teach about a full-scale nozzle only when the ratios controlling the dominant physics match. Dimensionless groups identify those ratios and prevent scaling by length alone.
Form Reynolds and Froude numbers
Inertial-to-viscous ratio is Re = ρUL/μ. For free surfaces, inertial-to-gravity ratio is Fr = U/√(gL). Matching Re preserves viscous regime; matching Fr preserves wave and gravity effects. A scale model cannot usually match every group at once, so choose according to the phenomenon being studied.
A 1:4 model has L_m = 0.25 L_p. To match Froude, U_m/U_p = √(L_m/L_p) = 0.5. To match Re in the same fluid, U_m/U_p = L_p/L_m = 4, an incompatibility that must be acknowledged.
Similarity is a qualified inference
If the pump, roughness, surface tension, and compressibility matter, additional Weber, relative roughness, or Mach numbers enter. A visually similar jet can still have wrong pressure loss.
Work it: For prototype U = 3 m/s, L = 0.4 m and water, compute Re and Fr. Choose model speed for Froude similarity at L = 0.05 m, then calculate the model Re and explain the mismatch.