Experimental investigation of transitional flow in a toroidal pipe

Abstract

The flow instability and further transition to turbulence in a toroidal pipe (torus) with curvature (tube-to-coiling diameter) 0.049 is investigated experimentally. The flow inside the toroidal pipe is driven by a steel sphere fitted to the inner pipe diameter. The sphere is moved with constant azimuthal velocity from outside the torus by a moving magnet. The experiment is designed to investigate curved pipe flow by optical measurement techniques. Using stereoscopic particle image velocimetry, laser Doppler velocimetry and pressure drop measurements the flow is measured for Reynolds numbers ranging from 1000 to 15 000. Time-and space-resolved velocity fields are obtained and analyzed. The steady axisymmetric basic flow is strongly influenced by centrifugal effects. On an increase of the Reynolds number we find a sequence of bifurcations. For Re = 4075 ± 2% a supercritical bifurcation to an oscillatory flow is found in which waves travel in streamwise direction with a phase velocity slightly faster than the mean flow. The oscillatory flow is superseded by a presumably quasi-periodic flow at a further increase of the Reynolds number before turbulence sets in. The results are found to be compatible, in general, with earlier experimental and numerical investigations on transition to turbulence in helical and curved pipes. However, important aspects of the bifurcation scenario differ considerably.

Modal instability of the flow in a toroidal pipe

“…All curvatures in the grid revealed spectra with unstable eigenvalues for Re < 6 × 10 3 . This result confirms the numerical observations by Di Piazza & Ciofalo (2011) as well as the experiments by Kühnen et al (2014Kühnen et al ( , 2015 and extends these studies, which are limited to the low curvature range. Most importantly, it proves that this flow is linearly unstable, in obvious contrast to the flow inside a straight pipe (see Schmid & Henningson (1994) and citations therein).…”

“…However, as indicated by Kühnen et al (2015) and as will be shown herein, these results should be considered inaccurate, with the exception of the symmetry characteristics observed in the flow. The only experiments employing toroidal pipes are those by Kühnen et al (2014Kühnen et al ( , 2015. These authors sacrificed the 2π (streamwise) periodicity and the mirror symmetry of the system by introducing a steel sphere in the tube to drive the fluid.…”

A comparative study on instability of steady flows in helical pipes

“…Comparing the calculated critical Reynolds numbers with the experimentally measured ones (Gelfgat, 2019), we obtained a good agreement with the recent experiments of Kühnen et al (2014Kühnen et al ( , 2015 done for < 0.1 and < 0.05. Several earlier experiments that studied the instability onset for similar pipe curvatures and small torsions (White, 1929;Taylor, 1929;Sreenivasan & Strykowski, 1983;Webster & Humphrey, 1993) found noticeably larger critical it can be estimated to be above 8000.…”

Critical Point for Bifurcation Cascades and Featureless Turbulence

“…Transition to turbulence is subcritical at low curvatures and qualitatively similar to the one in straight pipes [30][31][32]. For larger δ, instead, transition is initiated by a supercritical Hopf bifurcation [30,[32][33][34] and all elements point towards a bifurcation cascade [33]. The dynamics of the flow at intermediate curvatures, however, remains largely unexplored, leaving unanswered the question of whether the two transition scenarios interact, and if so how.…”

“…[30] but "could not be pinpointed" experimentally (quotations from Refs. [30,33]). Second, we move our attention to the region of the parameter space where the neutral curve of the flow intersects the threshold for subcritical transition [38], hereafter referred to as critical point, see Fig.…”