Numerical evidence of smooth self‐similar dynamics and possibility of subsequent collapse for three‐dimensional ideal flows (original) (raw)

Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations

Journal of Nonlinear Science, 2006

We study the interplay between the local geometric properties and the nonblowup of the 3D incompressible Euler equations. We consider the interaction of two perturbed antiparallel vortex tubes using Kerr's initial condition [15] [Phys. Fluids 5 (1993), 1725]. We use a pseudo-spectral method with resolution up to 1536×1024×3072 to resolve the nearly singular behavior of the Euler equations. Our numerical results demonstrate that the maximum vorticity does not grow faster than doubly exponential in time, up to t = 19, beyond the singularity time t = 18.7 predicted by Kerr's computations [15], [18]. The velocity, the enstrophy, and the enstrophy production rate remain bounded throughout the computations. As the flow evolves, the vortex tubes are flattened severely and turned into thin vortex sheets, which roll up subsequently. The vortex lines near the region of the maximum vorticity are relatively straight. This local geometric regularity of vortex lines seems to be responsible for the dynamic depletion of vortex stretching.

Velocity and scaling of collapsing Euler vortices

2006

New analysis of the scaling structure of a numerical solution of the Euler equations finds that initially anti-parallel vortex tubes collapse into two wings whose cross-sections can be described using two length scales ρ and R. The first ρ ∼ (T − t) for the leading edge and the distance between the position of peak vorticity and the dividing plane. The second R ∼ (T −t) 1/2 describes the extent of the wings and the distance of the peak in vortical velocity sup x |v| from the peak in vorticity. All measures of singular growth within the inner region give the same singular time. This includes a blowup in the peak of vortical or axial velocity going as (T − t) −1/2 at a distance R from the position of ω ∞ . Outside this self-similar region, energy, enstrophy, circulation and helicity accumulate. Twisting of vortex lines consistent with vortex line length growing to infinity is observed in the outer region. Vorticity in the intermediate zone between the inner and outer regions, while no longer growing at the singular rate, could be the major source of the strain interactions that drive the flow.

A model for the global structure of self-similar vortex sheet roll-up

2013

Most vortices are born from the roll-up of a shear layer. The roll-up is traditionally modeled as the self-similar winding of an infinite spiral connected to a thin shear layer: a vortex sheet. We demonstrate a composite vortex sheet-point vortex model to quantify the global structure of two archetypal cases of self-similar roll-up. These cases are Kaden’s single spiral solution of the wing-tip vortex and Pullin’s double spiral solution of the nonlinear impulse response of the Kelvin-Helmholtz instability. The model consists in replacing the spiral with a point vortex of equal vortical intensity and accounting for the mutual interaction of the point vortex and the untouched sheet. We show that the model itself has an attractive self-similar solution which compares well with numerical experiments.

Numerical study of singularity formation in a class of Euler and Navier-Stokes flows

We study numerically a class of stretched solutions of the three-dimensional Euler and Navier-Stokes equations identified by Gibbon, Fokas, and Doering ͑1999͒. Pseudo-spectral computations of a Euler flow starting from a simple smooth initial condition suggests a breakdown in finite time. Moreover, this singularity apparently persists in the Navier-Stokes case. Independent evidence for the existence of a singularity is given by a Taylor series expansion in time. The mechanism underlying the formation of this singularity is the two-dimensionalization of the vorticity vector under strong compression; that is, the intensification of the azimuthal components associated with the diminishing of the axial component. It is suggested that the hollowing of the vortex accompanying this phenomenon may have some relevance to studies in vortex breakdown.

Vortex events in Euler and Navier–Stokes simulations with smooth initial conditions

We present high-resolution numerical simulations of the Euler and Navier-Stokes equations for a pair of colliding dipoles. We study the possible approach to a finite-time singularity for the Euler equations, and contrast it with the formation of developed turbulence for the Navier-Stokes equations. We present numerical evidence that seems to suggest the existence of a blow-up of the inviscid velocity field at a finite time (t s ) with scaling |u| ∞ ∼ (t s − t) −1/2 , |ω| ∞ ∼ (t s − t) −1 . This blow-up is associated with the formation of a k −3 spectral range, at least for the finite range of wavenumbers that are resolved by our computation. In the evolution toward t s , the total enstrophy is observed to increase at a slower rate, Ω ∼ (t s − t) −3/4 , than would naively be expected given the behaviour of the maximum vorticity, ω ∞ ∼ (t s − t) −1 . This indicates that the blow-up would be concentrated in narrow regions of the flow field. We show that these regions have sheet-like structure. Viscous simulations, performed at various Re, support the conclusion that any non-zero viscosity prevents blow-up in finite time and results in the formation of a dissipative exponential range in a time interval around the estimated inviscid t s . In this case the total enstrophy saturates, and the energy spectrum becomes less steep, approaching k −5/3 . The simulations show that the peak value of the enstrophy scales as Re 3/2 , which is in accord with Kolmogorov phenomenology. During the short time interval leading to the formation of an inertial range, the total energy dissipation rate shows a clear tendency to become independent of Re, supporting the validity of Kolmogorov's law of finite energy dissipation. At later times the kinetic energy shows a t −1.2 decay for all Re, in agreement with experimental results for grid turbulence. Visualization of the vortical structures associated with the stages of vorticity amplification and saturation show that, prior to t s , large-scale and the small-scale vortical structures are well separated. This suggests that, during this stage, the energy transfer mechanism is non-local both in wavenumber and in physical space. On the other hand, as the spectrum becomes shallower and a k −5/3 range appears, the energy-containing eddies and the small-scale vortices tend to be concentrated in the same regions, and structures with a wide range of sizes are observed, suggesting that the formation of an inertial range is accompanied by transfer of energy that is local in both physical and spectral space.

Geometry and dynamics of vortex sheets in 3 dimension

Theoretical and Applied Mechanics, 2002

We consider the properties and dynamics of vortex sheets from a geometrical, coordinate-free, perspective. Distribution-valued forms (de Rham currents) are used to represent the fluid velocity and vorticity due to the vortex sheets. The smooth velocities on either side of the sheets are solved in terms of the sheet strengths using the language of double forms. The classical results regarding the continuity of the sheet normal component of the velocity and the conservation of vorticity are exposed in this setting. The formalism is then applied to the case of the self-induced velocity of an isolated vortex sheet. We develop a simplified expression for the sheet velocity in terms of representative curves. Its relevance to the classical Localized Induction Approximation (LIA) to vortex filament dynamics is discussed.

The collapse of an axi-symmetric, swirling vortex sheet

Nonlinearity, 1993

An ax-symmetric and swirling vortex sheet is investigated as the simplest flow in which there is non-trivial vortex stretching and as a possible setting for studying vortex cancellation and singularity formation. Rayleigh's criterion indicates linear stabiliry of a single sheet but instability for other configurations of sheets. Due to the simplicity of vortex sheet problems, the linear modes and growth rates (or frequencies) can be explicitly expressed. Subsequent nonlinear evolution is numerically simulated using a vortex method. The numerical results for an ai-symmetric swirling sheet with a vortex line along the axis of symmetry show detlchment of a vortex ring from the sheet into the outer fluid, and Follapse of the sheet onto t 2?c vortex line at some points. Vortex cancellation, which in the presence of viscosity would likely lead to vortex line reconnection. seems to occur in both of these phenomena. The evolution of two co-axial, axi-symmetric, swirling vortex sheets is similar.

Potential singularity mechanism for the Euler equations

Physical Review Fluids, 2016

Singular solutions to the Euler equations could provide essential insight on the formation of very small scales in highly turbulent flows. Previous attempts to find singular flow structures have proven inconclusive. We reconsider the problem of interacting vortex tubes, for which it has long been observed that the flattening of the vortices inhibits sustained self-amplification of velocity gradients. Here, we consider an iterative mechanism, based on the transformation of vortex filaments into sheets, and their subsequent instability back into filaments. Elementary fluid mechanical arguments are provided to support the formation of singular structure via this iterated mechanism, which we analyze based on a simplified model of filament interactions.

Viscous evolution of point vortex equilibria: The collinear state

Physics of Fluids, 2010

When point vortex equilibria of the 2D Euler equations are used as initial conditions for the corresponding Navier-Stokes equations (viscous), typically an interesting dynamical process unfolds at short and intermediate time scales, before the long time single peaked, self-similar Oseen vortex state dominates. In this paper, we describe the viscous evolution of a collinear three vortex structure that corresponds to an inviscid point vortex fixed equilibrium. Using a multi-Gaussian 'core-growth' type of model, we show that the system immediately begins to rotate unsteadily, a mechanism we attribute to a 'viscously induced' instability. We then examine in detail the qualitative and quantitative evolution of the system as it evolves toward the long-time asymptotic Lamb-Oseen state, showing the sequence of topological bifurcations that occur both in a fixed reference frame, and in an appropriately chosen rotating reference frame. The evolution of passive particles in this viscously evolving flow is shown and interpreted in relation to these evolving streamline patterns. 2 Problem Setting Consider an incompressible fluid in an unbounded two-dimensional (2D) domain R 2. The fluid motion is governed by Navier-Stokes equations, written in terms of the vorticity field ω(x, t), a scalar-valued function