Steady, fully developed turbulent flow of an incompressible fluid of density moves…

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

Fluid Dynamics, Ch Transition to turbulence

Problem

Steady, fully developed turbulent flow of an incompressible fluid of density ρ\rho moves along a straight horizontal pipe of diameter DD, over a length LL, at mean speed uˉ\bar u, with a pressure drop Δp\Delta p over that length. ((a)) By balancing the rate of work done by the pressure difference on the fluid occupying that length of pipe against the rate of viscous dissipation within it (the mean kinetic and potential energy of this fluid do not change along a fully developed, horizontal length), derive the dissipation rate per unit mass ε=ΔpL uˉρ.\varepsilon = \frac{\Delta p}{L}\,\frac{\bar u}{\rho}. ((b)) Supposing that the size ℓK\ell_K of the smallest, most rapidly dissipating eddies depends only on the kinematic viscosity ν\nu and on ε\varepsilon, use dimensional analysis to derive ℓK=(ν3/ε)1/4\ell_K = (\nu^3/\varepsilon)^{1/4}. ((c)) Water (ρ=998 kg m−3\rho = 998\,\mathrm{kg\,m^{-3}}, μ=1.00×10−3 Pa s\mu = 1.00\times10^{-3}\,\mathrm{Pa\,s}) flows at a mean speed of 2.0 m s−12.0\,\mathrm{m\,s^{-1}} along a smooth pipe of diameter 4.0 cm4.0\,\mathrm{cm}. Using the Blasius friction law fD=0.316 Re−1/4f_D = 0.316\,Re^{-1/4} (valid for 4000<Re<1054000 < Re < 10^5) and the Darcy-Weisbach relation Δp/L=fD ρuˉ2/(2D)\Delta p/L = f_D\,\rho\bar u^2/(2D), find ε\varepsilon and ℓK\ell_K for this flow.

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