scholarly journals A mechanism for linear instability in two-dimensional rimming flow

2002 ◽  
Vol 60 (2) ◽  
pp. 283-299 ◽  
Author(s):  
S. B. G. O’Brien
Keyword(s):  
Linear Instability ◽  
Two Dimensional ◽  
Rimming Flow

scholarly journals The magnetorotational instability prefers three dimensions

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190622
Author(s):  
Jeffrey S. Oishi ◽  
Geoffrey M. Vasil ◽  
Morgan Baxter ◽  
Andrew Swan ◽  
Keaton J. Burns ◽  
...  
Keyword(s):  
Shear Rate ◽  
Angular Velocity ◽  
Laboratory Experiments ◽  
Three Dimensional ◽  
Linear Instability ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Magnetorotational Instability ◽  
Dynamo Action ◽  
Wide Range

The magnetorotational instability (MRI) occurs when a weak magnetic field destabilizes a rotating, electrically conducting fluid with inwardly increasing angular velocity. The MRI is essential to astrophysical disc theory where the shear is typically Keplerian. Internal shear layers in stars may also be MRI-unstable, and they take a wide range of profiles, including near-critical. We show that the fastest growing modes of an ideal magnetofluid are three-dimensional provided the shear rate, S , is near the two-dimensional onset value, S c . For a Keplerian shear, three-dimensional modes are unstable above S  ≈ 0.10 S c , and dominate the two-dimensional modes until S  ≈ 2.05 S c . These three-dimensional modes dominate for shear profiles relevant to stars and at magnetic Prandtl numbers relevant to liquid-metal laboratory experiments. Significant numbers of rapidly growing three-dimensional modes remainy well past 2.05 S c . These finding are significant in three ways. First, weakly nonlinear theory suggests that the MRI saturates by pushing the shear rate to its critical value. This can happen for systems, such as stars and laboratory experiments, that can rearrange their angular velocity profiles. Second, the non-normal character and large transient growth of MRI modes should be important whenever three-dimensionality exists. Finally, three-dimensional growth suggests direct dynamo action driven from the linear instability.

On the stability of Poiseuille flow of a Bingham fluid

1994 ◽  
Vol 263 ◽  
pp. 133-150 ◽  
Author(s):  
I. A. Frigaard ◽  
S. D. Howison ◽  
I. J. Sobey
Keyword(s):  
Yield Stress ◽  
Poiseuille Flow ◽  
Linear Instability ◽  
Bingham Fluid ◽  
Additional Parameter ◽  
Two Dimensional ◽  
Plane Poiseuille Flow ◽  
Bingham Number ◽  
The Stability ◽  
Drilling Muds

The stability to linearized two-dimensional disturbances of plane Poiseuille flow of a Bingham fluid is considered. Bingham fluids exhibit a yield stress in addition to a plastic viscosity and this description is typically applied to drilling muds. A non-zero yield stress results in an additional parameter, a Bingham number, and it is found that the minimum Reynolds number for linear instability increases almost linearly with increasing Bingham number.

Dynamics of vorticity defects in stratified shear flow

2012 ◽  
Vol 694 ◽  
pp. 292-331 ◽  
Author(s):  
N. J. Balmforth ◽  
A. Roy ◽  
C. P. Caulfield
Keyword(s):  
Linear Instability ◽  
Shear Instability ◽  
Two Dimensional ◽  
Computationally Efficient ◽  
Stratified Shear Flow ◽  
Prandtl Numbers ◽  
High Reynolds ◽  
Efficient Exploration ◽  
Secondary Instabilities ◽  
Sharp Interfaces

AbstractWe consider the linear stability and nonlinear evolution of two-dimensional shear flows that take the form of an unstratified plane Couette flow that is seeded with a localized ‘defect’ containing sharp density and vorticity variations. For such flows, matched asymptotic expansions furnish a reduced model that allows a straightforward and computationally efficient exploration of flows at sufficiently high Reynolds and Péclet numbers that sharp density and vorticity gradients persist throughout the onset, growth and saturation of instability. We are thereby able to study the linear and nonlinear dynamics of three canonical variants of stratified shear instability: Kelvin–Helmholtz instability, the Holmboe instability, and the lesser-considered Taylor instability, all of which are often interpreted in terms of the interactions of waves riding on sharp interfaces of density and vorticity. The dynamics near onset is catalogued; if the interfaces are sufficiently sharp, the onset of instability is subcritical, with a nonlinear state existing below the linear instability threshold. Beyond onset, both Holmboe and Taylor instabilities are susceptible to inherently two-dimensional secondary instabilities that lead to wave mergers and wavelength coarsening. Additional two-dimensional secondary instabilities are also found to appear for higher Prandtl numbers that take the form of parasitic Holmboe-like waves.

scholarly journals Instabilities of the flow around a cylinder and emission of vortex dipoles

2013 ◽  
Vol 730 ◽  
pp. 419-441 ◽  
Author(s):  
Ziv Kizner ◽  
Viacheslav Makarov ◽  
Leon Kamp ◽  
GertJan van Heijst
Keyword(s):  
Linear Stability ◽  
Linear Instability ◽  
Instability Region ◽  
Base Flow ◽  
Two Dimensional ◽  
Azimuthal Mode ◽  
Dimensional Parameter ◽  
Contour Dynamics ◽  
Instability Regions ◽  
Dynamics Simulations

AbstractInstabilities and long-term evolution of two-dimensional circular flows around a rigid circular cylinder (island) are studied analytically and numerically. For that we consider a base flow consisting of two concentric neighbouring rings of uniform but different vorticity, with the inner ring touching the cylinder. We first study the inviscid linear stability of such flows to perturbations of the free edges of the rings. For a given ratio of the vorticity in the rings, the governing parameters of the problem are the radii of the inner and outer rings scaled on the cylinder radius. In this two-dimensional parameter space, we determine analytically the regions of linear stability/instability of each azimuthal mode $m= 1, 2, \ldots . $ In the physically most meaningful case of zero net circulation, for each mode $m\gt 1$, two regions are identified: a regular instability region where mode $m$ is unstable along with some other modes, and a unique instability region where only mode $m$ is unstable. After the conditions of linear instability are established, inviscid contour-dynamics and high-Reynolds-number finite-element simulations are conducted. In the regular instability regions, simulations of both kinds typically result in the formation of vortical dipoles or multipoles. In the unique instability regions, where the inner vorticity ring is much thinner than the outer ring, the inviscid contour-dynamics simulations do not reveal dipole emission. In the viscous simulation, because viscosity has time to widen the inner ring, the instability develops in the same manner as in the regular instability regions.

The influence of viscoelasticity on the existence of steady solutions in two-dimensional rimming flow

1992 ◽  
Vol 235 (-1) ◽  
pp. 611 ◽  
Author(s):  
Dilip Rajagopalan ◽  
Ronald J. Phillips ◽  
Robert C. Armstrong ◽  
Robert A. Brown ◽  
Arijit Bose
Keyword(s):  
Two Dimensional ◽  
Rimming Flow ◽  
Steady Solutions

scholarly journals Kelvin–Helmholtz billows above Richardson number

2019 ◽  
Vol 879 ◽  
Author(s):  
J. P. Parker ◽  
C. P. Caulfield ◽  
R. R. Kerswell
Keyword(s):  
Dynamical System ◽  
Richardson Number ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Finite Width ◽  
Simple Explanation ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Hyperbolic Tangent ◽  
Uniform Background

We study the dynamical system of a two-dimensional, forced, stratified mixing layer at finite Reynolds number $Re$, and Prandtl number $Pr=1$. We consider a hyperbolic tangent background velocity profile in the two cases of hyperbolic tangent and uniform background buoyancy stratifications, in a domain of fixed, finite width and height. The system is forced in such a way that these background profiles are a steady solution of the governing equations. As is well known, if the minimum gradient Richardson number of the flow, $Ri_{m}$, is less than a certain critical value $Ri_{c}$, the flow is linearly unstable to Kelvin–Helmholtz instability in both cases. Using Newton–Krylov iteration, we find steady, two-dimensional, finite-amplitude elliptical vortex structures – i.e. ‘Kelvin–Helmholtz billows’ – existing above $Ri_{c}$. Bifurcation diagrams are produced using branch continuation, and we explore how these diagrams change with varying $Re$. In particular, when $Re$ is sufficiently high we find that finite-amplitude Kelvin–Helmholtz billows exist when $Ri_{m}>1/4$ for the background flow, which is linearly stable by the Miles–Howard theorem. For the uniform background stratification, we give a simple explanation of the dynamical system, showing the dynamics can be understood on a two-dimensional manifold embedded in state space, and demonstrate the cases in which the system is bistable. In the case of a hyperbolic tangent stratification, we also describe a new, slow-growing, linear instability of the background profiles at finite $Re$, which complicates the dynamics.

Direct numerical simulations of hypersonic boundary-layer transition with finite-rate chemistry

2014 ◽  
Vol 755 ◽  
pp. 35-49 ◽  
Author(s):  
Olaf Marxen ◽  
Gianluca Iaccarino ◽  
Thierry E. Magin
Keyword(s):  
Large Amplitude ◽  
High Speed ◽  
Nonlinear Effects ◽  
Linear Instability ◽  
Boundary Layer Transition ◽  
Nonlinear Instability ◽  
Primary Wave ◽  
Two Dimensional ◽  
Finite Rate ◽  
Layer Transition

AbstractThe paper describes a numerical investigation of linear and nonlinear instability in high-speed boundary layers. Both a frozen gas and a finite-rate chemically reacting gas are considered. The weakly nonlinear instability in the presence of a large-amplitude two-dimensional wave is investigated for the case of fundamental resonance. Depending on the amplitude of this two-dimensional primary wave, strong growth of oblique secondary perturbations occurs for favourable relative phase differences between the two. For essentially the same primary amplitude, secondary amplification is almost identical for a reacting and a frozen gas. Therefore, chemical reactions do not directly affect the growth of secondary perturbations, but only indirectly through the change of linear instability and hence amplitude of the primary wave. When the secondary disturbances reach a sufficiently large amplitude, strongly nonlinear effects stabilize both primary and secondary perturbations.

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