A long horizontal pipe of constant cross-section and length m leads from a large open…
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 Potential flow, complex potentials and lift
Problem
A long horizontal pipe of constant cross-section and length L=25 m leads from a large open reservoir and discharges to the atmosphere through a valve at its far end. The water surface in the reservoir stands H=4.0 m above the pipe axis and is kept at that level; the pipe axis and the outlet are at the same height, which is taken as the datum z=0. The water has density ρ=1000kgm−3, g=9.8ms−2, and the flow is treated as incompressible, inviscid and irrotational. Because the pipe has constant area, the speed u(t) is the same at every point along it; because the reservoir is large, its own contribution to the unsteady term is negligible, so along the streamline from the reservoir surface to the outlet ∫∂u/∂tds=Ldu/dt. The valve is opened fully at t=0 with the water at rest.
((a)) Use the unsteady Bernoulli equation between the reservoir surface and the outlet to find the acceleration of the water immediately after the valve is opened.
((b)) Find the steady speed the flow tends to.
((c)) Estimate the time scale of the start-up by asking how long the initial acceleration would need to reach that steady speed.
((d)) A Pitot tube facing upstream inside the pipe measures the difference between stagnation and static pressure once the flow is steady. What difference does it read?