A straight horizontal wind-tunnel duct carries air of density kg m

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

Fluids and Continuum Physics, Ch Bernoulli at work: Venturi meters, Pitot tubes and lift

Problem

A straight horizontal wind-tunnel duct carries air of density 1.201.20 kg m−3^{-3}, treated as incompressible, in steady inviscid flow. Upstream the cross-sectional area is 0.200.20 m2^{2}; the duct then contracts smoothly to a throat of area 0.0800.080 m2^{2}. A Pitot-static tube on the axis of the upstream section reads a difference of 300300 Pa between its stagnation hole and its static holes. Take the velocity as uniform over each cross-section. ((a)) Find the upstream speed on the axis. ((b)) Using the steady integral mass balance on a control volume between the upstream section and the throat, find the speed at the throat. ((c)) Along the axis of this horizontal duct, Euler's equation for steady flow reduces to ρ u dudx=−dpdx\rho\,u\,\dfrac{\mathrm{d}u}{\mathrm{d}x} = -\dfrac{\mathrm{d}p}{\mathrm{d}x}, so the pressure gradient balances the convective acceleration alone. Integrate this along the axis from the upstream section to the throat and evaluate the pressure drop.

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