Hyperbolic Stagnation Points Research Articles

Nested separatrices in simple shear flows: The effect of localized disturbances on stagnation lines

The effects of localized two-dimensional disturbances on the structure of shear flows featuring a stagnation line are investigated. A simple superposition of a planar Couette flow and Moffatt’s [J. Fluid Mech. 18, 1–18 (1964)] streamfunction for the decay of a disturbance between infinite stationary parallel plates shows that in general the stagnation line is replaced by a chain of alternating elliptic and hyperbolic stagnation points with a separation equal to 2.78 times the half-gap between the plates. The flow structure associated with each saddle point consists of a homoclinic separatrix and two other separatrices which locally diverge but become parallel far from the disturbance. This basic structure repeats to give a sequence of nested separatrices permitting the streamfunction to approach that of simple shear flow far from the disturbance. Using the finite-element method, the specific disturbance caused by a stationary cylinder placed on the stagnation line is considered, and results confirm the existence of the stagnation point chain, with computed separations and velocity damping ratios in very good agreement with those obtained from the Couette-Moffatt superposition. Numerical solutions also illustrate that while Reynolds number greatly affects the stagnation point separation and velocity damping ratio, these two quantities are the same for any pair of adjacent stagnation points in a given chain. Insight gained from the analysis of planar shear flows is applied to the flow in a half-filled horizontal annulus between rotating coaxial cylinders, and is used to explain why only certain flow patterns from the range of mathematically possible structures arise in previous numerical solutions. By way of contrast, the concentric annulus solution is then perturbed to allow for a small eccentricity. The nonuniformity of the intercylinder gap is shown to destroy the chain of stagnation points, but also to unfold additional flow structures not realizable when the gap is uniform.

Long time evolution of quantum averages near stationary points

We construct explicit expressions for quantum averages in coherent states for a Hamiltonian of degree 4 with a hyperbolic stagnation point. These expressions are valid for all times and “collapse” (i.e., become infinite) along a discrete sequence of times. We compute quantum corrections compared to classical expressions. These corrections become significant over a time period of order C log 1 ℏ .

Enhanced stretching of material lines by antiparallel vortex pairs in turbulence

Deformations of material lines in homogeneous isotropic turbulence of an incompressible viscous fluid are numerically investigated to understand the physical mechanism of the enhanced stretching of material lines in turbulence. It is shown qualitatively by careful visualizations of numerical data and quantitatively by detailed numerical analysis that tubular vortical coherent structures in small scales of turbulence play crucial roles in such intensive stretching of material lines. In particular, the tubular vortices tend to align to each other in an antiparallel manner, and a pair of antiparallel vortices strongly stretch material lines in two regions around the hyperbolic stagnation points between them.

Chaotic Advection of the Shallow Kuroshio Coastal Waters

The shallow coastal water of the Enshu-Nada Sea (ENSW) recirculates regardless of whether the Kuroshio path is straight or has meanders. The recirculation is formed as a result of flow separation at the sharp coastline. The outputs of a recent numerical simulation of the Kuroshio current, including case of a short-term meander caused by an anticyclonic eddy, were analysed to track the motion of the ENSW. In contrast to the steady-flow cases, the unsteady cases showed that the ENSW discharges into the Kuroshio Extension region and intrudes further north into the Kuroshio-Oyashio confluence region due to chaotic advection. Two hyperbolic stagnation points of the velocity field characterise the transport paths; one south of the Izu peninsula and the other at the Kuroshio Extension. This mechanism exists even without the Ekman drift and may play an important role in the transportation of the fish eggs and larvae from the southern Japan spawning ground to the food abundant Kuroshio-Oyashio transition zone.

The nonlinear development of three-dimensional disturbances at hyperbolic stagnation points: A model of the braid region in mixing layers

The properties of steady, two-dimensional flows with spatially uniform strain rates ε and rotation rates γ where ε2⩾γ2, and hence open, hyperbolic, streamlines are investigated. By comparison with a high resolution numerical simulation of a free shear layer, such a quadratic flow is an idealized local model of the “braid” region which develops between neighboring saturated Kelvin–Helmholtz billows in an unstable free shear layer. A class of exact three-dimensional nonlinear solutions for spatially periodic perturbations is derived. These solutions satisfy the condition that the amplitude of the time-varying wave number of the perturbation remains bounded in time, and hence that pressure plays an asymptotically small role in their dynamics. In the limit of long time, the energy of such perturbations in an inviscid flow grows exponentially, with growth rate 2ε2−γ2, and the perturbation pressure plays no significant role in the dynamic evolution. This asymptotic growth rate is not the maximal growth rate accessible to general perturbations, which may grow transiently at rate 2ε, independently of γ. However, almost all initial conditions lead to, at most, transient growth and hence finite asymptotic perturbation energy in an inviscid flow as time increases, due to the finite amplitude effects of pressure perturbations. Perturbations which do undergo significant transient growth take the form of streamwise-aligned perturbation vorticity which varies periodically in the spanwise direction. By comparison of this local model with a numerically simulated mixing layer, appropriately initialized “hyperbolic instabilities” appear to have significantly larger transient growth rates than an “elliptical instability” of the primary billow core. These hyperbolic instabilities appear to be a simple model for the spanwise periodic perturbations which are known to lead to the nucleation of secondary rib vortices in the braid region between adjacent billow cores.

An illustration of the link between ribs and hyperbolic instability

The hyperbolic instability in planar Taylor–Green cells is investigated by direct numerical simulations at Reynolds number 400. The mechanism of this three-dimensional instability is a stretching of the vorticity perturbation along the principal axis of strain in accordance with well-known results of rapid distortion theory and stability analysis, leading to the formation of spanwise counter-rotating vortices (or ribs) in the irrotational hyperbolic stagnation point regions, as described by Kerr and Dold [J. Fluid Mech. 276, 307 (1994)].

Effects of the Coriolis force on the stability of Stuart vortices

A detailed investigation of the effects of the Coriolis force on the three-dimensional linear instabilities of Stuart vortices is proposed. This exact inviscid solution describes an array of co-rotating vortices embedded in a shear flow. When the axis of rotation is perpendicular to the plane of the basic flow, the stability analysis consists of an eigenvalue problem for non-parallel versions of the coupled Orr–Sommerfeld and Squire equations, which is solved numerically by a spectral method. The Coriolis force acts on instabilities as a ‘tuner’, when compared to the non-rotating case. A weak anticyclonic rotation is destabilizing: three-dimensional Floquet modes are promoted, and at large spanwise wavenumber their behaviour is predicted by a ‘pressureless’ analysis. This latter analysis, which has been extensively discussed for simple flows in a recent paper (Leblanc & Cambon 1997) is shown to be relevant to the present study. The basic mechanism of short-wave breakdown is a competition between instabilities generated by the elliptical cores of the vortices and by the hyperbolic stagnation points in the braids, in accordance with predictions from the ‘geometrical optics’ stability theory. On the other hand, cyclonic or stronger anticyclonic rotation kills three-dimensional instabilities by a cut-off in the spanwise wavenumber. Under rapid rotation, the Stuart vortices are stabilized, whereas inertial waves propagate.

On the two-way interaction in two-dimensional particle-laden flows: the accumulation of particles and flow modification

The evolution of two-dimensional regular flows laden with solid heavy particles is studied analytically and numerically. The particulate phase is assumed to be dilute enough to neglect the effects of particle-particle interactions. Flows with large Reynolds and Froude numbers are considered, when effects related to viscous dissipation and gravity are negligible. A Cauchy problem is solved for an initially uniform distribution of particles with Stokes (St) and Reynolds (Rep) numbers of order unity in several types of flows representing steady solutions of the two-dimensional Euler equations. We consider flows in the vicinity of the hyperbolic stagnation point (with a uniform strain and zero vorticity) and the elliptic stagnation point (where vorticity is uniform), a circular vortex (with vorticity depending on the radius) and Stuart vortex flow. Analytical solutions are obtained, for the case of sufficiently small St, describing the accumulation of particles and corresponding modification of the fluid flow. Solutions derived show that the concentration of particles, although remaining uniform, decreases at the elliptic stagnation point and grows at the hyperbolic point. Owing to the coupling between the particulate and fluid dynamics, the flow vorticity is reduced at the elliptic point, while flow strain rate is enhanced at the hyperbolic point. Solutions obtained for the circular vortex show that the accumulation of particles proceeds in the form of a travelling wave. The concentration grows locally, forming the crest of the wave which propagates away from the vortex centre. Owing to the influence of the particulate on the carrier flow, the vorticity is reduced in the vortex centre. At the location of the crest the gradient of the flow grows owing to the drag forces between the fluid and particles and a vorticity peak is generated. Analytical solutions are also obtained for a chain of particle-laden Stuart vortices. Owing to the coupling effects, the concentration is diminished and the vorticity is reduced at the centres of the vortices. A sheet of increased concentration and vorticity is formed extending from the braid region to the periphery of the vortices, and the flow strain in the braid region is enhanced. Results of numerical simulations performed for St = 0.5 show good agreement with analytical solutions.

On the two-way interaction in two-dimensional particle-laden flows: the accumulation of particles and flow modification

The evolution of two-dimensional regular flows laden with solid heavy particles is studied analytically and numerically. The particulate phase is assumed to be dilute enough to neglect the effects of particle-particle interactions. Flows with large Reynolds and Froude numbers are considered, when effects related to viscous dissipation and gravity are negligible. A Cauchy problem is solved for an initially uniform distribution of particles with Stokes (St) and Reynolds (Rep) numbers of order unity in several types of flows representing steady solutions of the two-dimensional Euler equations. We consider flows in the vicinity of the hyperbolic stagnation point (with a uniform strain and zero vorticity) and the elliptic stagnation point (where vorticity is uniform), a circular vortex (with vorticity depending on the radius) and Stuart vortex flow. Analytical solutions are obtained, for the case of sufficiently small St, describing the accumulation of particles and corresponding modification of the fluid flow. Solutions derived show that the concentration of particles, although remaining uniform, decreases at the elliptic stagnation point and grows at the hyperbolic point. Owing to the coupling between the particulate and fluid dynamics, the flow vorticity is reduced at the elliptic point, while flow strain rate is enhanced at the hyperbolic point. Solutions obtained for the circular vortex show that the accumulation of particles proceeds in the form of a travelling wave. The concentration grows locally, forming the crest of the wave which propagates away from the vortex centre. Owing to the influence of the particulate on the carrier flow, the vorticity is reduced in the vortex centre. At the location of the crest the gradient of the flow grows owing to the drag forces between the fluid and particles and a vorticity peak is generated. Analytical solutions are also obtained for a chain of particle-laden Stuart vortices. Owing to the coupling effects, the concentration is diminished and the vorticity is reduced at the centres of the vortices. A sheet of increased concentration and vorticity is formed extending from the braid region to the periphery of the vortices, and the flow strain in the braid region is enhanced. Results of numerical simulations performed for St = 0.5 show good agreement with analytical solutions.

Hydrodynamic instability for certain ABC flows

Abstract A sufficient condition for instability of the Euler equations [see (1.5)] is used to demonstrate the instability of certain ABC flows. In the parameter range A = 1, B 2 + C 2 > 1, instability follows from the presence of hyperbolic stagnation points. In the parameter range A = 1, B = C = «e1, instability follows from the existence of hyperbolic closed trajectories and the associated exponential stretching of the fluid particles. This result is proved analytically in Section 2 and illustrated numerically via Poincare sections in Section 3.

Magnetic field evolution in steady chaotic flows

The fast dynamo problem for steady chaotic flows is approached by evolving magnetic field numerically at zero magnetic diffusion. Two flows are used, an ABC flow and a model flow constructed by patching simple flows together; both flows possess chaotic webs. From smooth initial conditions the magnetic field evolves fine structure in the chaotic regions of the flow because of exponential stretching and folding. Evidence for constructive folding of magnetic field within the chaotic regions is obtained: spatial averages of the magnetic field grow exponentially in time and this process is robust to different methods of averaging. This constructive folding is suggestive of fast dynamo action since the main effect of weak diffusion is to average field locally and smooth out fine variations. The growth and behaviour of initial conditions belonging to different symmetry classes is explored. In the case of the model flow, the dynamo process is explained in terms of chaotic stretching and folding together with cutting and pasting at hyperbolic stagnation points.

Dynamo theory, vorticity generation, and exponential stretching.

A discussion is given of the analogy between the dynamo equation for the generation of a magnetic field by the motion of an electrically conducting fluid and the equation for the evolution of vorticity of a viscous fluid. In both cases exponential stretching is an important feature of the underlying instability problem. For the "fast" dynamo problem, the existence of exponential stretching (i.e., the positivity of the Lyapunov exponent) somewhere in the flow is a necessary condition when the flow is smooth. An example is presented of a flow with exponential stretching (an Anosov flow) that supports fast dynamo action. A parallel treatment is described for the linearized Navier-Stokes equations for the motion of a viscous fluid. In this problem the analogous necessary condition for "fast vorticity generation" is the existence of some instability in the corresponding Euler (i.e., inviscid) equation. Dynamo theory methods give a second related result, namely a universal geometric estimate from below on the growth rate of a small perturbation in an inviscid fluid. This bound gives an effective sufficient condition for local instability for Eulers equations. In particular, it is proved that a steady flow with a hyperbolic stagnation point is unstable. The growth rate of an infinitesimal perturbation in a metric with derivatives depends on this metric. This dependence is completely described.

Instability criteria for the flow of an inviscid incompressible fluid.

We present a geometric estimate from below on the growth rate of a small perturbation of a three-dimensional steady flow of an ideal fluid and thus we obtain effective criteria for local instability for Euler's equations. We use these criteria to demonstrate the instability of several simple flows and to show that any flow with a hyperbolic stagnation point is unstable.