A flat roof drains through a vertical downpipe of internal diameter

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Viscous flow and dynamical similarity

Fluids and Continuum Physics, Ch Dimensional analysis, dynamical similarity and the dimensionless groups

Problem

A flat roof drains through a vertical downpipe of internal diameter d=50 mmd = 50\ \mathrm{mm}. In a heavy shower the pipe runs completely full of water and carries a steady volume flow rate Q=0.60 L s−1Q = 0.60\ \mathrm{L\,s^{-1}}. Take the water to be incompressible with density ρ=1000 kg m−3\rho = 1000\ \mathrm{kg\,m^{-3}} and dynamic viscosity μ=1.0×10−3 Pa s\mu = 1.0\times10^{-3}\ \mathrm{Pa\,s}, and take the flow to be laminar below Re≈2300Re \approx 2300 and turbulent above about 40004000, with ReRe built on the internal diameter and the mean speed. ((a)) Use conservation of mass for the full pipe to find the mean speed of the water. ((b)) Evaluate the Reynolds number of the pipe flow. ((c)) Is the flow laminar or turbulent? Name the dimensionless group that decided it and the two forces it compares.

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