Air ( , ) flows at a mean speed through a long smooth circular duct of diameter

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Instability, transition and turbulence · Instability, transition and turbulence

Fluid Dynamics, Ch Transition to turbulence

Problem

Air (ρ=1.20 kg m−3\rho = 1.20\,\mathrm{kg\,m^{-3}}, ν=1.51×10−5 m2 s−1\nu = 1.51\times10^{-5}\,\mathrm{m^2\,s^{-1}}) flows at a mean speed uˉ=5 m s−1\bar{u} = 5\,\mathrm{m\,s^{-1}} through a long smooth circular duct of diameter D=0.10 mD = 0.10\,\mathrm{m}. Treat the flow as steady, fully developed and turbulent, and use the Blasius friction law fD=0.316 Re−1/4f_D = 0.316\,Re^{-1/4}. ((a)) Find the Reynolds number and the Darcy friction factor fDf_D. ((b)) Find the pressure gradient Δp/L\Delta p/L and the dissipation per unit mass ε=(Δp/L)uˉ/ρ\varepsilon = (\Delta p/L)\bar{u}/\rho. ((c)) Estimate the Kolmogorov scale ℓK=(ν3/ε)1/4\ell_K = (\nu^3/\varepsilon)^{1/4}, and comment on whether a hot-wire probe of size 100 μm100\,\mu\mathrm{m} could resolve the smallest eddies.

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