On the stability of high-Reynolds-number flows with closed streamlines

1989 ◽  
Vol 200 ◽  
pp. 19-38 ◽  
Author(s):  
A. J. Mestel
Keyword(s):  
Rotating Magnetic Field ◽  
Two Dimensional ◽  
Inviscid Flows ◽  
Stability Theorems ◽  
High Reynolds ◽  
The Stability ◽  
Two Parameters ◽  
Reynolds Number Flows ◽  
Flow Reversals ◽  
Entire Theory

In steady, two-dimensional, inertia-dominated flows it is well known that the vorticity is constant along the streamlines, which, in a bounded domain, are necessarily closed. For inviscid flows, the variation of vorticity across the streamlines is arbitrary, while for forced, weakly dissipitative flows, it is determined by the balance between viscous diffusion and the forcing. This paper discusses the linear stability of flows of this type to two-dimensional disturbances. Arnol'd's stability theorems are discussed. An alternative functional to Arnol'd's is found, which gives the same stability criteria and which permits a representation of the problem in terms of a Schrödinger equation. Conditions for stability are derived from this functional. In particular it is shown that total flow reversals are potentially unstable. The results are illustrated with respect to the geometrically simple case when the streamlines are circular and the forcing is due to a rotating magnetic field, for which case the stability regions are calculated as a function of two parameters. It is shown that the entire theory, including Arnol'd's theorems, applies also to poloidal axisymmetric flows.

scholarly journals Bifurcation Tracking for High Reynolds Number Flow Around an Airfoil

2017 ◽  
Vol 27 (04) ◽  
pp. 1750061 ◽  
Author(s):  
S. Huntley ◽  
D. Jones ◽  
A. Gaitonde
Keyword(s):  
Reynolds Number ◽  
High Reynolds Number ◽  
Stability Boundary ◽  
Small Scale ◽  
Geometrical Parameters ◽  
Reynolds Number Flow ◽  
High Reynolds ◽  
The Stability ◽  
Two Parameters ◽  
Reynolds Number Flows

High Reynolds number flows are typical for many applications including those found in aerospace. In these conditions nonlinearities arise which can, under certain conditions, result in instabilities of the flow. The accurate prediction of these instabilities is vital to enhance understanding and aid in the design process. The stability boundary can be traced by following the path of a bifurcation as two parameters are varied using a direct bifurcation tracking method. Historically, these methods have been applied to small-scale systems and only more recently have been used for large systems as found in Computational Fluid Dynamics. However, these have all been concerned with flows that are inviscid. We show how direct bifurcation tracking methods can be applied efficiently to high Reynolds number flows around an airfoil. This has been demonstrated through the presentation of a number of test cases using both flow and geometrical parameters.

scholarly journals Complex Canard Explosion in a Fractional-Order FitzHugh–Nagumo Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950111 ◽  
Author(s):  
Mohammed-Salah Abdelouahab ◽  
René Lozi ◽  
Guanrong Chen
Keyword(s):  
Dynamical System ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Search Algorithm ◽  
Two Dimensional ◽  
Mixed Mode Oscillations ◽  
The Stability ◽  
Two Parameters ◽  
Complex Phenomena ◽  
Autonomous Dynamical System

This article investigates the complex phenomena of canard explosion with mixed-mode oscillations, observed from a fractional-order FitzHugh–Nagumo (FFHN) model. To rigorously analyze the dynamics of the FFHN model, a new mathematical notion, referred to as Hopf-like bifurcation (HLB), is introduced. HLB provides a precise definition for the change between a fixed point and an [Formula: see text]-asymptotically [Formula: see text]-periodic solution of the fractional-order dynamical system, as well as the stability of the FFHN model and the appearance of the HLB. The existence of canard oscillations in the neighborhoods of such HLB points are numerically investigated. Using a new algorithm, referred to as the global-local canard explosion search algorithm, the appearance of various patterns of solutions is revealed, with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied. The numbers of such oscillations versus the two parameters, respectively, are perfectly fitted using exponential functions. Finally, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional order close to one. After all, the article demonstrates that the FFHN model is a very simple two-dimensional model with an incredible ability to present the complex dynamics of neurons.

scholarly journals The stability theorems for discrete dynamical systems on two-dimensional manifolds

1983 ◽  
Vol 90 ◽  
pp. 1-55 ◽  
Author(s):  
Atsuro Sannami
Keyword(s):  
Dynamical Systems ◽  
Smooth Manifold ◽  
Riemannian Metric ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Stability Theorems ◽  
The Stability

One of the basic problems in the theory of dynamical systems is the characterization of stable systems.Let M be a closed (i.e. compact without boundary) connected smooth manifold with a smooth Riemannian metric and Diffr (M) (r ≥ 1) denote the space of Cr diffeomorphisms on M with the uniform Cr topology.

Global stability of two-dimensional and axisymmetric Euler flows

1994 ◽  
Vol 276 ◽  
pp. 273-305 ◽  
Author(s):  
P. A. Davidson
Keyword(s):  
Magnetic Relaxation ◽  
Two Dimensional ◽  
Inviscid Flows ◽  
Wide Range ◽  
Recirculating Flows ◽  
Close Relationship ◽  
Euler Flows ◽  
Relaxation Procedure ◽  
The Stability ◽  
Planar Flows

This paper is concerned with the stability of steady inviscid flows with closed streamlines. In increasing order of complexity we look at two-dimensional planar flows, poloidal (r, z) flows, and swirling recirculating flows. In each case we examine the relationship between Arnol’d's variational approach to stability, Moffatt's magnetic relaxation technique, and a more recent relaxation procedure developed by Valliset al.We start with two-dimensional (x, y) flows. Here we show that Moffatt's relaxation procedure will, under a wide range of circumstances, produce Euler flows which are stable. The physical reasons for this are discussed in the context of the well-known membrane analogy. We also show that there is a close relationship between Hamilton's principle and magnetic relaxation. Next, we examine poloidal flows. Here we find that, by and large, our planar results also hold true for axisymmetric flows. In particular, magnetic relaxation once again provides stable Euler flows. Finally, we consider swirling recirculating flows. It transpires that the introduction of swirl has a profound effect on stability. In particular, the flows produced by magnetic relaxation are no longer stable. Indeed, we show that all swirling recirculating Euler flows are potentially unstable to the extent that they fail to satisfy Arnol’d's stability criterion. This is, perhaps, not surprising, as all swirling recirculating flows include regions where the angular momentum decreases with radius and we would intuitively expect such flows to be prone to a centrifugal instability. The paper concludes with a discussion of marginally unstable modes in swirling flows. In particular, we examine the extent to which Rayleigh's original ideas on stability may be generalized, through the use of the Routhian, to include flows with a non-zero recirculation.

Stability of two-dimensional collapsible-channel flow at high Reynolds number

2012 ◽  
Vol 705 ◽  
pp. 371-386 ◽  
Author(s):  
Ramesh B. Kudenatti ◽  
N. M. Bujurke ◽  
T. J. Pedley
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Poiseuille Flow ◽  
Pressure Difference ◽  
High Reynolds Number ◽  
Boundary Layer Theory ◽  
Unexpected Finding ◽  
Two Dimensional ◽  
High Reynolds ◽  
The Stability

AbstractWe study the linear stability of two-dimensional high-Reynolds-number flow in a rigid parallel-sided channel, of which part of one wall has been replaced by a flexible membrane under longitudinal tension ${T}^{\ensuremath{\ast} } $. Far upstream the flow is parallel Poiseuille flow at Reynolds number $\mathit{Re}$; the width of the channel is $a$ and the length of the membrane is $\lambda a$, where $1\ll {\mathit{Re}}^{1/ 7} \lesssim \lambda \ll \mathit{Re}$. Steady flow was studied using interactive boundary-layer theory by Guneratne & Pedley (J. Fluid Mech., vol. 569, 2006, pp. 151–184) for various values of the pressure difference ${P}_{e} $ across the membrane at its upstream end. Here unsteady interactive boundary-layer theory is used to investigate the stability of the trivial steady solution for ${P}_{e} = 0$. An unexpected finding is that the flow is always unstable, with a growth rate that increases with ${T}^{\ensuremath{\ast} } $. In other words, the stability problem is ill-posed. However, when the pressure difference is held fixed (${= }0$) at the downstream end of the membrane, or a little further downstream, the problem is well-posed and all solutions are stable. The physical mechanisms underlying these findings are explored using a simple inviscid model; the crucial factor in the fluid dynamics is the vorticity gradient across the incoming Poiseuille flow.

An iterative method for high-Reynolds-number flows with closed streamlines

1989 ◽  
Vol 200 ◽  
pp. 1-18 ◽  
Author(s):  
A. J. Mestel
Keyword(s):  
Iterative Method ◽  
Stream Function ◽  
Surface Stress ◽  
Stokes Equations ◽  
Functional Dependence ◽  
Tangential Velocity ◽  
Skin Depth ◽  
Rotating Magnetic Field ◽  
Inviscid Flows ◽  
High Reynolds

In steady, two-dimensional, inviscid flows it is well-known that, in the absence of rotational forcing, the vorticity is constant along streamlines. In a bounded domain the streamlines are necessarily closed. In some circumstances, investigated in this paper, this behaviour is exhbited also by forced viscous flows, when the variation of vorticity across the streamlines is determined by a balance between viscous diffusion and the forcing. Similar results hold in axisymmetry. For such flows, an iterative process for finding the vorticity as a function of the stream function is described. The method applies whenever the viscous boundary condition can be expressed in terms of the vorticity or tangential stress rather then the tangential velocity. When it is applicable, the iterative method is faster than direct solution of the Navier-Stokes equations at high Reynolds numbers. As an example, the method is used to calculate the flow in a model of the electromagnetic stirring process. In this model, a conducting fluid in an elliptical region is driven by a rotating magnetic field and resisted by a surface stress. The functional dependence of the vorticity on the stream function is found for various values of the magnetic skin depth, surface stress and eccentricity of the ellipse. The form of the flow is discussed with particular reference to whether it consists of a single circulatory region or separates into two or more such regions.

Modeling and Simulation of High Reynolds' Number Flows During Reaction Injection Mold Filling

1994 ◽  
Vol 9 (3) ◽  
pp. 279-285 ◽  
Author(s):  
Rahima K. Mohammed ◽  
Tim A. Osswald ◽  
Timothy J. Spiegelhoff ◽  
Esther M. Sun
Keyword(s):  
Reynolds Number ◽  
Modeling And Simulation ◽  
High Reynolds Number ◽  
Mold Filling ◽  
Injection Mold ◽  
High Reynolds ◽  
Reynolds Number Flows
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