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The motion of a disk in a Langmuir film bounding a liquid substrate is a classical hydrodynamic problem, dating back to Saffman (J. Fluid Mech., vol. 73, 1976, p. 593) who focused upon the singular problem of translation at large Boussinesq number,$${\textit {Bq}}\gg 1$$. A semianalytic solution of the dual integral equations governing the flow at arbitrary$${\textit {Bq}}$$was devised by Hugheset al.(J. Fluid Mech., vol. 110, 1981, p. 349). When degenerated to the inviscid-film limit$${\textit {Bq}}\to 0$$, it produces the value$$8$$for the dimensionless translational drag, which is$$50\,\%$$larger than the classical$$16/3$$-value corresponding to a free surface. While that enhancement has been attributed to surface incompressibility, the mathematical reasoning underlying the anomaly has never been fully elucidated. Here we address the inviscid limit$${\textit {Bq}}\to 0$$from the outset, revealing a singular mechanism where half of the drag is contributed by the surface pressure. We proceed beyond that limit, considering a nearly inviscid film. A naïve attempt to calculate the drag correction using the reciprocal theorem fails due to an edge singularity of the leading-order flow. We identify the formation of a boundary layer about the edge of the disk, where the flow is primarily in the azimuthal direction with surface and substrate stresses being asymptotically comparable. Utilising the reciprocal theorem in a fluid domain tailored to the asymptotic topology of the problem produces the drag correction$$(8\,{\textit {Bq}}/{\rm \pi} ) [ \ln (2/{\textit {Bq}}) + \gamma _E+1]$$,$$\gamma _E$$being the Euler–Mascheroni constant.
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Criteria for dynamic stall onset and vortex shedding in low-Reynolds-number flows
Dynamic stall at low Reynolds numbers,$$Re \sim O(10^4)$$, exhibits complex flow physics with co-existing laminar, transitional and turbulent flow regions. Current state-of-the-art stall onset criteria use parameters that rely on flow properties integrated around the leading edge. These include the leading edge suction parameter or$$LESP$$(Rameshet al.,J. Fluid Mech., vol. 751, 2014, pp. 500–538) and boundary enstrophy flux or$$BEF$$(Sudharsanet al.,J. Fluid Mech., vol. 935, 2022, A10), which have been found to be effective for predicting stall onset at moderate to high$$Re$$. However, low-$$Re$$flows feature strong vortex-shedding events occurring across the entire airfoil surface, including regions away from the leading edge, altering the flow field and influencing the onset of stall. In the present work, the ability of these stall criteria to effectively capture and localize these vortex-shedding events in space and time is investigated. High-resolution large-eddy simulations for an SD7003 airfoil undergoing a constant-rate, pitch-up motion at two$$Re$$(10 000 and 60 000) and two pitch rates reveal a rich variety of unsteady flow phenomena, including instabilities, transition, vortex formation, merging and shedding, which are described in detail. While stall onset is reflected in both$$LESP$$and$$BEF$$, local vortex-shedding events are identified only by the$$BEF$$. Therefore,$$BEF$$can be used to identify both dynamic stall onset and local vortex-shedding events in space and time.
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- Award ID(s):
- 1935255
- PAR ID:
- 10563391
- Publisher / Repository:
- Journal of Fluid Mechanics
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 996
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2024.753
