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Precessing vortex cores (PVC), arising from a global instability in swirling flows, can dramatically alter the dynamics of swirl-stabilized flames. Previous study of these instabilities has identified their frequencies and potential for interaction with the shear layer instabilities also present in swirling flows. In this work, we investigate the dynamics of precessing vortex cores at a range of swirl numbers and the impact that turbulence, which tends to increase with swirl number due to the increase in mean shear, has on the dynamics of this instability. This is particularly interesting as stability predictions have previously incorporated turbulence effects using an eddy viscosity model, which only captures the impact of turbulence on the base flow, not on the instantaneous dynamics of the PVC itself. Time-resolved experimental measurements of the three-component velocity field at ten swirl numbers show that at lower swirl numbers, the PVC is affected by turbulence through the presence of vortex jitter. With increasing swirl number, the PVC jitter decreases as the PVC strength increases. There is a critical swirl number below which jitter of the PVC vortex monotonically increases with increasing swirl number, and beyond which the jitter decreases, indicating that the strength of the PVC dominates over turbulent fluctuations at higher swirl numbers, despite the fact that the turbulence intensities continue to rise with increasing swirl number. Further, we use a nonlinear van der Pol oscillator model to explain the competition between the random turbulent fluctuations and coherent oscillations of the PVC. The results of this work indicate that while both the strength of the PVC and magnitude of turbulence intensity increase with increasing swirl number, there are defined regimes where each of them hold a stronger influence on the large-scale, coherent dynamics of the flow field.
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Crazy Sequential Representations of Numbers for Small Bases
Abstract Throughout history, recreational mathematics has always played a prominent role in advancing research. Following in this tradition, in this paper we extend some recent work with crazy sequential representations of numbers− equations made of sequences of one through nine (or nine through one) that evaluate to a number. All previous work on this type of puzzle has focused only on base ten numbers and whether a solution existed. We generalize this concept and examine how this extends to arbitrary bases, the ranges of possible numbers, the combinatorial challenge of finding the numbers, efficient algorithms, and some interesting patterns across any base. For the analysis, we focus on bases three through ten. Further, we outline several interesting mathematical and algorithmic complexity problems related to this area that have yet to be considered.
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- Award ID(s):
- 1817602
- PAR ID:
- 10178890
- Date Published:
- Journal Name:
- Recreational Mathematics Magazine
- Volume:
- 6
- Issue:
- 12
- ISSN:
- 2182-1976
- Page Range / eLocation ID:
- 33 to 48
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.2478/rmm-2019-0007
