Fluid Mechanics · Differential conservation laws and Navier-Stokes structure
A 1:25 model of a surface vessel is to be tested in a towing tank that can use either…
Problem
A 1:25 model of a surface vessel is to be tested in a towing tank that can use either freshwater (ρ = 998 kg/m³, μ = 1.00 × 10⁻³ Pa·s) or a glycerin–water mixture (ρ = 1150 kg/m³, μ = 0.020 Pa·s). The prototype operates in seawater (ρ = 1025 kg/m³, μ = 1.07 × 10⁻³ Pa·s) at V_p = 6.00 m/s with length ℓ_p = 50.0 m. Show that matching both the Reynolds number and the Froude number with the same model fluid is impossible, and compute the viscosity μ_req (Pa·s) that a hypothetical model fluid of density 998 kg/m³ would need in order to match both dimensionless numbers at the Froude-scaled speed.
Hint
Write the two required model speeds from Fr = V/√(gℓ) and Re = ρVℓ/μ.
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