Researchers quantify the appearance of turbulence in a pipe folded on itself

How much stress do pipes experience when liquid passes through them, and how does this depend on the degree of curvature of the pipe?

The bends of the pipes are particularly crucial, for example in the aortic arch which connects to the left ventricle of the human heart. Piping systems in industrial facilities often include 90 degree or greater elbows, may be helical, and may even have 180 degree elbows. Swedish fluid mechanics analyzed the flow of fluids in such pipes bent 180 degrees. Their research is published in the journal Physical Exam Fluids.

Pipe bends are different from their straight sections because in curved sections there are outward centrifugal forces due to the inertia of the liquid inside. This force is balanced by a pressure gradient from the outer wall of the pipe to the inner wall. Because the fluid velocities in an imaginary slice through the pipe will not be equal in the curved section (for example, the velocity near the outer wall of the pipe will be greater than near the inner wall), a model of Secondary flow, in addition to movement through the pipe, is installed perpendicular to the main flow direction.

This movement is a pair of counter-rotating symmetrical vortices, called Dean’s vortices, named after the British scientist William Reginald Dean, which appear in the first bend of the pipe and can complicate subsequent flow, both laminar and turbulent flow. .

For a single elbow, the internal geometry of the flow can be described by the Dean number, which depends on the radius of the pipe relative to the curvature of the elbow, and by the Reynolds number of the fluid, which is the ratio of forces of inertia. viscous forces within a fluid. Fluids have a critical Reynolds number that characterizes their transition from smooth laminar flow to turbulent flow, and this can be twice as large as in straight flow. (In fact, turbulent flow from a straight pipe can become laminar again when entering a spiral section of the pipe.)

Roughly speaking, Reynolds numbers below 2000 indicate laminar flow, those above 3500 indicate turbulent flow, with a transition from laminar to turbulent flow occurring somewhere in between. The Dean number measures the intensity of the secondary internal flow.

Daniele Massaro and colleagues at KTH Royal Institute of Technology in Stockholm used a refined method to solve the famous complex Navier-Stokes fluid equations numerically and computationally to analyze the transition (from laminar to turbulent flow). ) in an idealized pipe with a 180 degree elbow, comparing their results to previous results for elbow (90 degree elbow) and toroidal pipes.

Assuming a representative pipe curvature of 1/3 (the ratio of the radius of a pipe cross-section to the radius of curvature), the group divided the simulated fluid into about 30 million grids, not all of them uniform. They then solved the equations for the grid points as they changed over time.

By performing a stability analysis (determining the growth of tiny, infinitesimal imperfections that appear in the initial smooth fluid), the calculation determines changes in the fluid as it goes around the bend. Changes occur in all vertical sections of the fluid and along the entire length of the pipe. In this way, the transition from laminar to turbulent flow can be determined.

The intense calculation – for which supercomputers were needed, Massaro said, with runs that could take months – revealed that the critical Reynolds number for the transition was 2,528. This is the region of the number Reynolds of the fluid, regardless of its type, where instability appears and the shape of the structure leads to the transition to turbulence. This transition point is also known as the “Hopf bifurcation”. The instability of a 180 degree turn develops in the same way as that of a 90 degree turn. The critical Reynolds number for a 90 degree turn is 2531 and for a torus it is 3290.

Due to the detailed nature of the instability, pipes with bends greater than 180 degrees should be similar, to a certain extent. For pipes with shorter bends, the Hopf bifurcation should disappear as the bend angle approaches zero, with flow remaining laminar. The group estimates that the bifurcation disappears at a turn of about 20 degrees.

Although the research has obvious industrial applications, extending it to the heart is not straightforward due to the difference between real blood and the flow idealized by this study. “Our study helps to understand where a sudden transition in the laminar aortic arch, typically laminar, might occur,” said Massaro, study co-author and graduate student in the department of mechanical engineering at the Royal Institute of Medicine. KTH technology from Stockholm. . “Indeed, the turbulent regime in the aorta may potentially be linked to various heart diseases.”

© 2023 Science X Network