Flow Laws Fluid Mechanics

Demonstration of basic flow laws through flow measurements using the ultrasonic Doppler method

Flow measurements using the ultrasonic Doppler method demonstrate fundamental laws of fluid flow in pipes and their dependence on flow velocity and pipe geometry.

Keywords: Laminar and turbulent flow, continuity equation, Bernoulli equation, Hagen-Poiseuille law, flow velocity, flow resistance, pressure scales, static and dynamic pressure, viscosity

The test setup can be used to measure the Doppler frequency shift for different pump speeds on measuring sections with different pipe diameters. At the same time, the corresponding pressure drops can be determined using riser pipes. This allows a clear demonstration of the laws that apply to a liquid with laminar flow. From the flow speeds determined using the Doppler method, the pipe geometries and the measured pressure drops, flow rates, flow resistances and the dynamic viscosity of the Doppler liquid can be determined by applying the continuity equation, the Bernoulli equation and the Hagen-Poiseuille law. By calculating the Reynolds numbers for the various flow speeds and pipe diameters, it can be checked whether stationary laminar flow conditions prevailed during the measurements.

Schematic representation of the experimental setup

Flow resistances for different pipe diameters and flows

The corresponding flow can be calculated from the measured flow velocities and the respective cross-sectional areas. In this test setup, this is almost the same for all pipe diameters with the same settings of the centrifugal pump and thus satisfies the continuity equation. As a further result, the flow resistance R determined for different pipe diameters and for different flows is shown in the diagram below. This shows the strong dependence on the pipe radius r that is to be expected according to the Hagen-Poiseuille law: R ~ 1/r ⁴ .

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