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.
more »
« less
Complex singularity analysis for vortex layer flows
We study the evolution of a 2D vortex layer at high Reynolds number. Vortex layer flows are characterized by intense vorticity concentrated around a curve. In addition to their intrinsic interest, vortex layers are relevant configurations because they are regularizations of vortex sheets. In this paper, we consider vortex layers whose thickness is proportional to the square-root of the viscosity. We investigate the typical roll-up process, showing that crucial phases in the initial flow evolution are the formation of stagnation points and recirculation regions. Stretching and folding characterizes the following stage of the dynamics, and we relate these events to the growth of the palinstrophy. The formation of an inner vorticity core, with vorticity intensity growing to infinity for larger Reynolds number, is the final phase of the dynamics. We display the inner core's self-similar structure, with the scale factor depending on the Reynolds number. We reveal the presence of complex singularities in the solutions of Navier–Stokes equations; these singularities approach the real axis with increasing Reynolds number. The comparison between these singularities and the Birkhoff–Rott singularity seems to suggest that vortex layers, in the limit $Re\rightarrow \infty$ , behave differently from vortex sheets.
more »
« less
- Award ID(s):
- 1928930
- NSF-PAR ID:
- 10347886
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 932
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)The instability and transition to turbulence and its evolution in pulsatile flows, which involve reverse flows and unsteady flow separations, is the primary focus of this experimental work. A piston driven by a programmable DC servo motor was used to set-up a water flow system and provide the pulsation characteristics. Time-resolved particle image velocimetry data were acquired in a refractive index matching set-up by using a continuous wave laser and a high-frame-rate digital camera. The position of the piston was continuously recorded by a laser proximity sensor. Five different experiments were carried out with Reynolds numbers in the range of 535–4825 and Womersley numbers from 11.91 to 23.82. The non-stationarity of the data was addressed by incorporating trend removal methods involving low- and high-pass filtering of the data, and using empirical mode decomposition together with the relevant Hilbert–Huang transform to determine the intrinsic mode functions. This latter method is more appropriate for nonlinear and non-stationary cases, for which traditional analysis involving classical Fourier decomposition is not directly applicable. It was found that transition to turbulence is a spontaneous event covering the whole near-wall region. The instantaneous vorticity profiles show the development of a large-scale ring-like attached wall vortical layer (WVL) with smaller vortices of higher frequencies than the pulsation frequency superimposed, which point to a shear layer Kelvin–Helmholtz (K–H) type of instability. Inflectional instability leads to flow separation and the formation of a major roll-up structure with the K–H vortices superimposed. This structure breaks down in the azimuthal direction into smaller turbulence patches with vortical content, which appears to be the prevailing structural content of the flow at each investigated Reynolds number ( Re ). At higher Re numbers, the strength and extent of the vortices are larger and substantial disturbances appear in the free stream region of the flow, which are typical of pipe flows at transitional Re numbers. Turbulence appears to be produced at the locations of maximum or minimum vorticity within the attached WVL, in the ridges between the K–H vortices around the separated WVL and the upstream side of the secondary vortex where the flow impinges on the wall. This wall turbulence breaks away into the middle section of the pipe, at approximately $Re \ge 2200$ , by strong eruptions of the K–H vortices.more » « less
-
The present work uses a reduced-order model to study the motion of a buoyant vortex ring with non-negligible core size. Buoyancy is considered in both non-Boussinesq and Boussinesq situations using an axisymmetric contour dynamics formulation. The density of the vortex ring differs from that of the ambient fluid, and both densities are constant and conserved. The motion of the ring is calculated by following the boundary of the vortex core, which is also the interface between the two densities. The velocity of the contour comes from a combination of a specific continuous vorticity distribution within its core and a vortex sheet on the core boundary. An evolution equation for the vortex sheet is derived from the Euler equation, which simplifies considerably in the Boussinesq limit. Numerical solutions for the coupled integro-differential equations are obtained. The dynamics of the vortex sheet and the formation of two possible singularities, including singularities in the curvature and the shock-like profile of the vortex sheet strength, are discussed. Three dimensionless groups, the Atwood, Froude and Weber numbers, are introduced to measure the importance of physical effects acting on the motion of a buoyant vortex ring.more » « less
-
In this study, the standard unsteady vortex lattice method (UVLM) is modified by different approaches to accommodate the viscous effects on the flow field and lift force history of a moving airfoil. Firstly, the conventional inviscid vortices are replaced with ones that satisfy Oseen’s approximation of the Navier–Stokes equations. These are known as viscous vortices and are a function of the Reynolds number. Compared to the standard UVLM, the results show a different behavior for the wake deformation at lower Reynolds numbers where the viscous effects are more dominant. The second modification is implemented by an elegant alteration in the conservation of circulation condition based on the fact that the whole domain composed of the fluid and solid can be treated as a single kinematic entity. The no-slip boundary condition enables considering the moving solid vorticity in the Kelvin condition, where the results for the frequency response indicate more phase lag compared to the standard UVLM or potential flow theories (i.e., Theodorsen), which increases at higher reduced frequencies. Furthermore, the effect of the position of the shed vortices is analyzed and seen to affect the amplitude of the resulting lift history. Finally, the no-slip boundary condition is also explicitly considered by implementing source singularities along the surface of a thick airfoil, resulting in a better prediction of the lift magnitude compared to CFD results. The combination of all the aforementioned modifications has the potential to enhance the performance of the UVLM in case of viscous flows at relatively high reduced frequencies.more » « less
-
In this paper a model for viscous boundary and shear layers in three dimensions is introduced and termed a vortex-entrainment sheet. The vorticity in the layer is accounted for by a conventional vortex sheet. The mass and momentum in the layer are represented by a two-dimensional surface having its own internal tangential flow. Namely, the sheet has a mass density per-unit-area making it dynamically distinct from the surrounding outer fluid and allowing the sheet to support a pressure jump. The mechanism of entrainment is represented by a discontinuity in the normal component of the velocity across the sheet. The velocity field induced by the vortex-entrainment sheet is given by a generalized Birkhoff–Rott equation with a complex sheet strength. The model was applied to the case of separation at a sharp edge. No supplementary Kutta condition in the form of a singularity removal is required as the flow remains bounded through an appropriate balance of normal momentum with the pressure jump across the sheet. A pressure jump at the edge results in the generation of new vorticity. The shedding angle is dictated by the normal impulse of the intrinsic flow inside the bound sheets as they merge to form the free sheet. When there is zero entrainment everywhere the model reduces to the conventional vortex sheet with no mass. Consequently, the pressure jump must be zero and the shedding angle must be tangential so that the sheet simply convects off the wedge face. Lastly, the vortex-entrainment sheet model is demonstrated on several example problems.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2021.966
