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The motion of a freely rotating prolate spheroid in a simple shear flow of a dilute polymeric solution is examined in the limit of large particle aspect ratio,$$\kappa$$. A regular perturbation expansion in the polymer concentration,$$c$$, a generalized reciprocal theorem, and slender body theory to represent the velocity field of a Newtonian fluid around the spheroid are used to obtain the$$O(c)$$correction to the particle's orientational dynamics. The resulting dynamical system predicts a range of orientational behaviours qualitatively dependent upon$$c\, De$$($$De$$is the imposed shear rate times the polymer relaxation time) and$$\kappa$$and quantitatively on$$c$$. At a small but finite$$c\, De$$, the particle spirals towards a limit cycle near the vorticity axis for all initial conditions. Upon increasing$$\kappa$$, the limit cycle becomes smaller. Thus, ultimately the particle undergoes a periodic motion around and at a small angle from the vorticity axis. At moderate$$c\, De$$, a particle starting near the flow–gradient plane departs it monotonically instead of spirally, as this plane (a limit cycle at smaller$$c\, De$$) obtains a saddle and an unstable node. The former is close to the flow direction. Upon further increasing$$c\, De$$, the saddle node changes to a stable node. Therefore, depending upon the initial condition, a particle may either approach a periodic orbit near the vorticity axis or obtain a stable orientation near the flow direction. Upon further increasing$$c\, De$$, the limit cycle near the vorticity axis vanishes, and the particle aligns with the flow direction for all starting orientations.
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This content will become publicly available on November 25, 2025
Impact of spanwise rotation on flow separation and recovery behind a bulge in channel flows
Direct numerical simulations of spanwise-rotating turbulent channel flow with a parabolic bump on the bottom wall are employed to investigate the effects of rotation on flow separation. Four rotation rates,$$Ro_b := 2\varOmega H/U_b = \pm 0.42$$,$$\pm$$1.0, are compared with the non-rotating scenario. The mild adverse pressure gradient induced by the lee side of the bump allows for a variable pressure-induced separation. The separation region is reduced (increased) when the bump is on the anti-cyclonic (cyclonic) side of the channel, compared with the non-rotating separation. The total drag is reduced in all rotating cases. Through several mechanisms, rotation alters the onset of separation, reattachment and wake recovery. The mean momentum deficit is found to be the key. A physical interpretation of the ratio between the system rotation and mean shear vorticity,$$S:=\varOmega /\varOmega _s$$, provides the mechanisms regarding stability thresholds$$S=-0.5$$and$$-$$1. The rotation effects are explained accordingly, with reference to the dynamics of several flow structures. For anti-cyclonic separation, particularly, the interaction between the Taylor–Görtler vortices and hairpin vortices of wall-bounded turbulence is proven to be responsible for the breakdown of the separating shear layer. A generalized argument is made regarding the essential role of near-wall deceleration and resultant ejection of enhanced hairpin vortices in destabilizing an anti-cyclonic flow. This mechanism is anticipated to have broad impacts on other applications in analogy to rotating shear flows, such as thermal convection and boundary layers over concave walls.
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- Award ID(s):
- 2131942
- PAR ID:
- 10589951
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 999
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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