Water ( , ) flows steadily at through a smooth pipe of diameter
PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Instability, transition and turbulence
Fluid Dynamics, Ch Transition to turbulence
Problem
Water (ρ=998kgm−3, ν=1.00×10−6m2s−1) flows steadily at uˉ1=3ms−1 through a smooth pipe of diameter D1=1.0cm, which then widens suddenly to a coaxial pipe of diameter D2=2.0cm. Take the Borda-Carnot energy loss per unit mass for a sudden expansion, (uˉ1−uˉ2)2/2, as given from a control-volume energy balance across the expansion (an application of the Bernoulli and momentum theorems for a control volume), and take the mixing region over which this energy is dissipated as extending about 8D2 downstream of the expansion.
((a)) Find the Reynolds number of the flow upstream of the expansion. Say whether an ordinary inlet would give laminar or turbulent flow there, and why a very quiet inlet could change that answer.
((b)) Find uˉ2 from mass conservation, the Borda-Carnot loss per unit mass, and the total power dissipated in the mixing region.
((c)) Estimate the dissipation per unit mass ε in the mixing region and the Kolmogorov scale ℓK=(ν3/ε)1/4 there.