Stationary longitudinal vortical rolls emerge in katabatic and anabatic Prandtl slope flows at shallow slopes as a result of an instability when the imposed surface buoyancy flux relative to the background stratification is sufficiently large. Here, we identify the self-pairing of these longitudinal rolls as a unique flow structure. The topology of the counter-rotating vortex pair bears a striking resemblance to speaker-wires and their interaction with each other is a precursor to further destabilization and breakdown of the flow field into smaller structures. On its own, a speaker-wire vortex retains its unique topology without any vortex reconnection or breakup. For a fixed slope angle $\alpha =3^{\circ }$ and at a constant Prandtl number, we analyse the saturated state of speaker-wire vortices and perform a bi-global linear stability analysis based on their stationary state. We establish the existence of both fundamental and subharmonic secondary instabilities depending on the circulation and transverse wavelength of the base state of speaker-wire vortices. The dominance of subharmonic modes relative to the fundamental mode helps to explain the relative stability of a single vortex pair compared to the vortex dynamics in the presence of two or an even number of pairs. These instability modes are essential for the bending and merging of multiple speaker-wire vortices, which break up and lead to more dynamically unstable states, eventually paving the way for transition towards turbulence. This process is demonstrated via three-dimensional flow simulations with which we are able to track the nonlinear temporal evolution of these instabilities.
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Cascades and reconnection in interacting vortex filaments
At high Reynolds number, the interaction between two vortex tubes leads to intense
velocity gradients, which are at the heart of fluid turbulence. This vorticity amplification
comes about through two different instability mechanisms of the initial vortex tubes, assumed
anti-parallel and with a mirror plane of symmetry. At moderate Reynolds number, the
tubes destabilize via a Crow instability, with the nonlinear development leading to strong
flattening of the cores into thin sheets. These sheets then break down into filaments which
can repeat the process. At higher Reynolds number, the instability proceeds via the elliptical
instability, producing vortex tubes that are perpendicular to the original tube directions. In
this work, we demonstrate that these same transition between Crow and Elliptical instability
occurs at moderate Reynolds number when we vary the initial angle between two straight
vortex tubes. We demonstrate that when the angle between the two tubes is close to =2,
the interaction between tubes leads to the formation of thin vortex sheets. The subsequent
breakdown of these sheets involves a twisting of the paired sheets, followed by the appearance
of a localized cloud of small scale vortex structures. At smaller values of the angle between
the two tubes, the breakdown mechanism changes to an elliptic cascade-like mechanism.
Whereas the interaction of two vortices depends on the initial condition, the rapid formation
of fine-scales vortex structures appears to be a robust feature, possibly universal at very high
Reynolds numbers.
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- Award ID(s):
- 2011754
- NSF-PAR ID:
- 10499981
- Publisher / Repository:
- American Physical Society
- Date Published:
- Journal Name:
- Physical Review Fluids
- Volume:
- 6
- Issue:
- 7
- ISSN:
- 2469-990X
- 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/10.1103/PhysRevFluids.6.074701
