On the transition to instability for compressible vortex sheets

2004 ◽  
Vol 134 (5) ◽  
pp. 885-892 ◽  
Author(s):  
Jean-François Coulombel ◽  
Paolo Secchi
Keyword(s):  
Linear Stability ◽  
Energy Estimate ◽  
Vortex Sheet ◽  
Limit Case ◽  
Vortex Sheets ◽  
Unstable Regime ◽  
Stable Regime ◽  
Weakly Stable

We study the linear stability of a vortex sheet in a limit case that corresponds to a transition between a weakly stable regime and a violently unstable regime. We prove an energy estimate that reflects the high degeneracy of the uniform Kreiss–Lopatinskii condition.

scholarly journals Anisotropic regularity of linearized compressible vortex sheets

2020 ◽  
Vol 17 (03) ◽  
pp. 443-458
Author(s):  
Paolo Secchi
Keyword(s):  
Weighted Sobolev Spaces ◽  
Vortex Sheet ◽  
Vortex Sheets ◽  
Frequency Space ◽  
Inviscid Fluids ◽  
Classical Stability ◽  
Pseudo Differential Equation ◽  
Constant Coefficients ◽  
Two Space Dimensions ◽  
Weakly Stable

We consider supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients, Morando et al. recently derived a pseudo-differential equation that describes the time evolution of the discontinuity front of the vortex sheet. In agreement with the classical stability analysis, the problem is weakly stable if [Formula: see text], and the well-posedness holds in standard weighted Sobolev spaces. Our aim in this paper is to improve this result, by showing the existence in functional spaces with additional weighted anisotropic regularity in the frequency space.

scholarly journals Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

2021 ◽  
Author(s):  
Javier Gómez-Serrano ◽  
Jaemin Park ◽  
Jia Shi ◽  
Yao Yao
Keyword(s):  
Angular Velocity ◽  
Disjoint Union ◽  
Vortex Sheet ◽  
Vortex Sheets ◽  
Smooth Curves ◽  
Rigidity Results ◽  
Positive Vorticity ◽  
Constant Vorticity ◽  
Concentric Circles ◽  
Finite Disjoint Union

AbstractIn this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $$\Omega $$ Ω , such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.

scholarly journals Collapse of n Point Vortices, Formation of the Vortex Sheets and Transport of Passive Markers

Energies ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 943
Author(s):  
Henryk Kudela
Keyword(s):  
Differential Equations ◽  
Point Vortex ◽  
Optimization Procedure ◽  
Vortex Sheet ◽  
Point Vortices ◽  
Vortex Sheets ◽  
Impermeable Barrier ◽  
Initial Location ◽  
Passive Tracers ◽  
Initial Iterate

In this paper, the motion of the n-vortex system as it collapses to a point in finite time is studied. The motion of vortices is described by the set of ordinary differential equations that we are able to solve analytically. The explicit formula for the solution demands the initial location of collapsing vortices. To find the collapsing locations of vortices, the algebraic, nonlinear system of equations was built. The solution of that algebraic system was obtained using Newton’s procedure. A good initial iterate needs to be provided to succeed in the application of Newton’s procedure. An unconstrained Leverber–Marquart optimization procedure was used to find such a good initial iterate. The numerical studies were conducted, and numerical evidence was presented that if in a collapsing system n=50 point vortices include a few vortices with much greater intensities than the others in the set, the vortices with weaker intensities organize themselves onto the vortex sheet. The collapsing locations depend on the value of the Hamiltonian. By changing the Hamiltonian values in a specific interval, the collapsing curves can be obtained. All points on the collapse curves with the same Hamiltonian value represent one collapsing system of vortices. To show the properties of vortex sheets created by vortices, the passive tracers were used. Advection of tracers by the velocity induced by vortices was calculated by solving the proper differential equations. The vortex sheets are an impermeable barrier to inward and outward fluxes of tracers. Arising vortex structures are able to transport the passive tracers. In this paper, several examples showing the diversity of collapsing structures with the vortex sheet are presented. The collapsing phenomenon of many vortices, their ability to self organize and the transportation of the passive tracers are novelties in the context of point vortex dynamics.

scholarly journals Confined vortex surface and irreversibility. 1. Properties of exact solution

2021 ◽  
pp. 2150097
Author(s):  
Alexander Migdal
Keyword(s):  
Strain Tensor ◽  
Special Kind ◽  
Local Strain ◽  
Vortex Sheet ◽  
Exceptional Case ◽  
Surface Integral ◽  
Time Reversibility ◽  
Navier Stokes Equation ◽  
Vortex Sheets ◽  
Asymmetric Vortex

We revise the steady vortex surface theory following the recent finding of asymmetric vortex sheets (Migdal, 2021). These surfaces avoid the Kelvin–Helmholtz instability by adjusting their discontinuity and shape. The vorticity collapses to the sheet only in an exceptional case considered long ago by Burgers and Townsend, where it decays as a Gaussian on both sides of the sheet. In generic asymmetric vortex sheets (Shariff, 2021), vorticity leaks to one side or another, making such sheets inadequate for vortex sheet statistics and anomalous dissipation. We conjecture that the vorticity in a turbulent flow collapses on a special kind of surface (confined vortex surface or CVS), satisfying some equations involving the tangent components of the local strain tensor. The most important qualitative observation is that the inequality needed for this solution’s stability breaks the Euler dynamics’ time reversibility. We interpret this as dynamic irreversibility. We have also represented the enstrophy as a surface integral, conserved in the Navier–Stokes equation in the turbulent limit, with vortex stretching and viscous diffusion terms exactly canceling each other on the CVS surfaces. We have studied the CVS equations for the cylindrical vortex surface for an arbitrary constant background strain with two different eigenvalues. This equation reduces to a particular version of the stationary Birkhoff–Rott equation for the 2D flow with an extra nonanalytic term. We study some general properties of this equation and reduce its solution to a fixed point of a map on a sphere, guaranteed to exist by the Brouwer theorem.

A numerical investigation of the rolling-up of vortex sheets

1980 ◽  
Vol 373 (1752) ◽  
pp. 67-91 ◽  
Keyword(s):  
Numerical Integration ◽  
Numerical Investigation ◽  
Equations Of Motion ◽  
Additional Term ◽  
Vortex Sheet ◽  
Time Step ◽  
Vortex Sheets ◽  
Numerical Instabilities ◽  
Sinusoidal Disturbance ◽  
Induced Velocity

In this paper the development of a vortex sheet due to an initially sinusoidal disturbance is calculated. When determining the induced velocity in points of the vortex sheet, it can be represented by concentrated vortices but it is shown that it is analytically more correct to add an additional term that represents the effect of the immediate neighbourhood of the point considered. The equations of motion were integrated by a Runge-Kutta technique to exclude numerical instabilities. The time step was determined by the requirement that a quantity (Hamiltonian) that remains invariant as a result of the equations of motion, should not change more than a certain amount in the numerical integration of the equations of motion. One difficulty is that if a greater number of concentrated vortices are introduced to represent the vortex sheet, the effect of round-off errors becomes more important. The number of figures retained in the computations limits the number of concentrated vortices. Where the round-off errors have been kept sufficiently small, a process of rolling-up of vorticity clearly occurs. There is no point in pursuing the calculations much beyond this point, first because the representation of the vortex sheet by concentrated vortices becomes more and more inaccurate and secondly because viscosity will have the effect of transforming the rolled-up vortex sheet into a region of vorticity.

Paper 7. Unsteady Flow around a Slender Fish-Like Body

1972 ◽  
Vol 14 (7) ◽  
pp. 43-52 ◽  
Author(s):  
Th. Y. Wu ◽  
J. N. Newman
Keyword(s):  
Unsteady Flow ◽  
Theoretical Calculations ◽  
Cross Flow ◽  
Vortex Sheet ◽  
The Body ◽  
Dynamic Force ◽  
Cross Sectional ◽  
Vortex Sheets ◽  
Flat Fish ◽  
The Cross

This paper attempts to extend some recent theoretical calculations on the unsteady flow generated by body movements of a slender ‘flat’ fish by further including the effect of finite body thickness in the consideration for various configurations of side and caudal fins as major appendages. Based on the slender-body approximation, the cross-flow is determined for different longitudinal body sections which are characterized by a variety of cross-sectional shapes and flow conditions (such as having smooth or fin-edged body contours, with or without vortex sheets alongside the body section). The effect of body thickness is found to arise primarily from its interaction with the vortex sheet already existing in the cross-flow. New results for the transverse hydro-dynamic force acting on the body are obtained, and their physical significances are discussed.

Fine Structure of Vortex Sheet Rollup by Viscous and Inviscid Simulation

1991 ◽  
Vol 113 (1) ◽  
pp. 31-36 ◽  
Author(s):  
G. Tryggvason ◽  
W. J. A. Dahm ◽  
K. Sbeih
Keyword(s):  
Reynolds Number ◽  
High Reynolds Number ◽  
Large Scale ◽  
Stokes Equations ◽  
Flow Region ◽  
Vortex Sheet ◽  
Navier Stokes ◽  
Vortex Sheets ◽  
Limiting Behavior ◽  
High Reynolds

Numerical simulations of the large amplitude stage of the Kelvin-Helmholtz instability of a relatively thin vorticity layer are discussed. At high Reynolds number, the effect of viscosity is commonly neglected and the thin layer is modeled as a vortex sheet separating one potential flow region from another. Since such vortex sheets are susceptible to a short wavelength instability, as well as singularity formation, it is necessary to provide an artificial “regularization” for long time calculations. We examine the effect of this regularization by comparing vortex sheet calculations with fully viscous finite difference calculations of the Navier-Stokes equations. In particular, we compare the limiting behavior of the viscous simulations for high Reynolds numbers and small initial layer thickness with the limiting solution for the roll-up of an inviscid vortex sheet. Results show that the inviscid regularization effectively reproduces many of the features associated with the thickness of viscous vorticity layers with increasing Reynolds number, though the simplified dynamics of the inviscid model allows it to accurately simulate only the large scale features of the vorticity field. Our results also show that the limiting solution of zero regularization for the inviscid model and high Reynolds number and zero initial thickness for the viscous simulations appear to be the same.

scholarly journals Nighttime wind and scalar variability within and above an Amazonian canopy

2018 ◽  
Vol 18 (5) ◽  
pp. 3083-3099 ◽  
Author(s):  
Pablo E. S. Oliveira ◽  
Otávio C. Acevedo ◽  
Matthias Sörgel ◽  
Anywhere Tsokankunku ◽  
Stefan Wolff ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Low Frequency ◽  
Amazon Forest ◽  
Stable Regime ◽  
Stable Conditions ◽  
Weakly Stable ◽  
The Stability ◽  
The Relationship ◽  
Temporal Windows ◽  
Scalar Variability

Abstract. Nocturnal turbulent kinetic energy (TKE) and fluxes of energy, CO2 and O3 between the Amazon forest and the atmosphere are evaluated for a 20-day campaign at the Amazon Tall Tower Observatory (ATTO) site. The distinction of these quantities between fully turbulent (weakly stable) and intermittent (very stable) nights is discussed. Spectral analysis indicates that low-frequency, nonturbulent fluctuations are responsible for a large portion of the variability observed on intermittent nights. In these conditions, the low-frequency exchange may dominate over the turbulent transfer. In particular, we show that within the canopy most of the exchange of CO2 and H2O happens on temporal scales longer than 100 s. At 80 m, on the other hand, the turbulent fluxes are almost absent in such very stable conditions, suggesting a boundary layer shallower than 80 m. The relationship between TKE and mean winds shows that the stable boundary layer switches from the very stable to the weakly stable regime during intermittent bursts of turbulence. In general, fluxes estimated with long temporal windows that account for low-frequency effects are more dependent on the stability over a deeper layer above the forest than they are on the stability between the top of the canopy and its interior, suggesting that low-frequency processes are controlled over a deeper layer above the forest.

On the kinetic theory of stable and weakly unstable plasma. Part 1

1968 ◽  
Vol 2 (1) ◽  
pp. 33-49 ◽  
Author(s):  
A. Rogister ◽  
C. Oberman
Keyword(s):  
Kinetic Theory ◽  
Thermal Contact ◽  
Stable State ◽  
Main Body ◽  
Secondary Importance ◽  
Stable Regime ◽  
Enhanced Diffusion ◽  
Weakly Stable ◽  
Stable Plasma ◽  
Unstable Plasma

A kinetic theory is presented which is valid for both weakly unstable and stable plasma. The theory corrects the conventional Balescu—Guernsey—Lenard description for the weakly stable portions of the fluctuation spectrum. The theory is no longer Markoffian in the distribution function F alone but is in the pair F and Ik the spectrum of fluctuations. Further the evolution in time from an initially weakly unstable distribution to the stable regime can be described. Even after passage into the stable state the fluctuations can be large nd lead to enhanced diffusion across a magnetic field. The coefficient of spatial diffusion is given in this weakly stable (or untable) state. A strong coupling is found between plasmons and the particles in the distribution with velocities near the phase velocities of the plasmons. There is only weak thermal contact between the plasmons and these particles with the main body of the distribution whenever certain nonlinear processes are of secondary importance.

Divergent Eddy Heat Fluxes in the Kuroshio Extension at 144°–148°E. Part II: Spatiotemporal Variability

2013 ◽  
Vol 43 (11) ◽  
pp. 2416-2431 ◽  
Author(s):  
Stuart P. Bishop
Keyword(s):  
Spatial Structure ◽  
Regime Shift ◽  
Kuroshio Extension ◽  
Spatiotemporal Variability ◽  
Unstable State ◽  
Unstable Regime ◽  
Stable Regime ◽  
Eddy Heat Flux ◽  
The Mean ◽  
The Kuroshio

Abstract The Kuroshio Extension System Study (KESS) provided 16 months of observations to quantify divergent eddy heat flux (DEHF) from a mesoscale-resolving array of current- and pressure-equipped inverted echo sounders. KESS observations captured a regime shift from a stable to unstable state. There is a distinct difference in the spatial structure of DEHFs between the two regimes. The stable regime had weak downgradient DEHFs. The unstable regime exhibited asymmetry along the mean path with strong downgradient DEHFs upstream of a mean trough at ~147°E. The spatial structure of DEHFs resulted from episodic mesoscale processes. The first 6 months were during the stable regime in which fluxes were associated with eastward-propagating 10–15-day upper meanders. After 6 months, the Kuroshio Extension underwent a regime shift from a stable to unstable state. This regime shift corresponded with a red shift in mesoscale phenomena with the prevalence of ~40-day deep externally generated eddies. DEHF amplitudes more than quadrupled during the unstable regime. Cold-core ring (CCR) formation, CCR–jet interaction, and coupling between ~40-day deep eddies were responsible for asymmetry in downgradient fluxes in the mean maps not observed during the stable regime. The Kuroshio Extension has prominent deep energy associated with externally generated eddies that interact with the jet to drive some of the biggest DEHF events. These eddies play an important role in the variability of the jet through eddy–mean flow interactions. The DEHFs that result from vertical coupling act in accordance with baroclinic instability. The interaction is not growth from an infinitesimal perturbation, but from the start is a finite-amplitude interaction.

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