Circular Streamlines Research Articles

Strained vortices in driven cavities

Vortices are of fundamental importance in fluid mechanics. An interesting aspect of vortex dynamics is the transition from two- to three-dimensional flow. The transition to three-dimensional flow of a circular vortex can be caused by a weak two-dimensional non-axisymmetric strain field deforming the circular streamlines. After briefly reviewing the three-dimensional instability of strained vortices in unbounded domains, the effect on vortex stability of bounding walls of an enclosure will be investigated by exploring the 2D–3D transition in a driven cavity. This system, in which the flow is driven by the steady tangential motion of one or two facing walls, is a paradigm for closed flows. The rich multitude of instabilities and bifurcations is demonstrated and analogies to unstable unbounded vortices are established. Contrary to vortices in open flows, the amplitudes of three-dimensional perturbations in driven cavities typically saturate. Hence, stationary or time-periodic three-dimensional vortices can be realized with ease, facilitating detailed experimental investigations.

The local instability of steady astrophysical flows with non circular streamlines with application to differentially rotating disks with free eccentricity

We carry out a general study of the stability of astrophysical flows that appear steady in a uniformly rotating frame. Such a flow might correspond to a stellar pulsation mode or an accretion disk with a free global distortion giving it finite eccentricity. We consider perturbations arbitrarily localized in the neighbourhood of unperturbed fluid streamlines.When conditions do not vary around them, perturbations take the form of oscillatory inertial or gravity modes. However, when conditions do vary so that a circulating fluid element is subject to periodic variations, parametric instability may occur. For nearly circular streamlines, the dense spectra associated with inertial or gravity modes ensure that resonance conditions can always be satisfied when twice the period of circulation round a streamline falls within. We apply our formalism to a differentially rotating disk for which the streamlines are Keplerian ellipses, with free eccentricity up to 0.7, which do not precess in an inertial frame. We show that for small $e,$ the instability involves parametric excitation of two modes with azimuthal mode number differing by unity in magnitude which have a period of twice the period of variation as viewed from a circulating unperturbed fluid element. Instability persists over a widening range of wave numbers with increasing growth rates for larger eccentricities. The nonlinear outcome is studied in a follow up paper which indicates development of small scale subsonic turbulence.

Global numerical simulations of differentially rotating disks with free eccentricity

We study the nonlinear evolution of global m = 1 modes with low pattern speeds in a differentially rotating non magnetic disk, by means of two and three dimensional numerical simulations. The modes make disk streamlines eccentric with maximum eccentricity in the range 0.13−0.24. We found that long lived patterns corresponding to eccentricities ∼0.1 lasted for the duration of the simulations of ∼64 orbits at the disk outer boundary. They had slow retrograde precession with period ∼10 orbits at the outer boundary, in good agreement with that found from linear normal mode analysis. As expected from linear stability analysis, which leads one to expect a parametric instability associated with the non circular streamlines, we also found that three dimensional simulations showed the growth of a local instability to producing vertical motions on a local scale that eventually results in small amplitude turbulence with root mean square vertical velocity typically ∼0.03cs, cs being the sound speed. This turbulence together with much more effective shocks produced by the interaction of the eccentric disk with the inner boundary were responsible for damping the disk eccentricity. Our first estimate of the damping time associated with turbulence for eccentricities ∼0.1, corresponded to that associated with a turbulent viscosity acting to circularize the eccentric streamlines with α parameter ∼10 −3 . For parameters appropriate to protostellar disks at 5 AU this corresponds to a decay time ∼10 5−6 y. Thus although the indication is that disks with eccentric streamlines are long lived, there is associated turbulence even in the non magnetic case leading to estimated decay times that may be significant when the possibility of the growth of orbital eccentricity for extrasolar planets is considered through disk planet interaction on sufficiently long timescales.

Phenomenological Characterization of Vortex Induced Scour

A series of experiments, investigating vortex induced scour formation, show that the scour hole geometry induced by a rotating fluid flow, can be explained within the concept that the granular bed is eroded as the imposed shear stress overcomes a certain critical value inherent in the liquid/bed pair. The situation is seemingly kindred to that observed in the pseudoplastic Bingham fluids and the first experiment, investigating cylindrical scour, is conducted using a high viscous Bingham fluid with a low yield stress. Two other experiments investigating the two and three dimensional scour formation for the stationary flow regime are also conducted using a fluid and grain substrate pair. A simple model for vortex induced scour is derived for an infinitely long axisymmetric vortex with circular streamlines surrounded by a deformable boundary. The Navier–Stokes equations are solved by using a polynomial approximation of the tangential velocity. The model is then extended to two and three dimensional stationary flow situations by considering the torque balance between the fluid motion and the critical bed shear stress. The quantified results agree well with experimental results obtained in both the one dimensional Bingham fluid experiment and the two and three dimensional fluid/grain experiments.

Chaotic dynamics in a strained rotating flow: a precessing plane fluid layer

The nonlinear dynamics exhibited by a planar layer of precessing fluid is examined as a canonical example of a strained rotating flow. The simple basic flow, Ubasic = −YXˆ+(X−2εZ)Yˆ in a frame rotating at εXˆ, consists of sheared circular streamlines (where ε measures the shearing) which are linearly unstable through the pairwise resonance of two inertial waves in a fashion similar to elliptical flow. Direct numerical simulation shows that the weakly nonlinear regime is quickly disrupted by further instabilities which lead to a multitude of co-existing solution branches, some of which represent chaotic flows. All these solutions remain within O(ε) (in an energy norm) of Ubasic so that energy is not apparently withdrawn from the fluid's underlying rotation. Further increases in the precession rate cause the flow to branch-switch randomly between these now quasi-stable states so that a new form of ‘slow’ dynamics emerges. The implication of this and the fact that these instabilities can nevertheless be classed as ‘strong’ is discussed from the perspective of the closely related problem of the precessing Earth and laboratory models thereof.

Secondary instabilities of rotating flows in the presence of a magnetic field: The case of circular streamlines

The stability of standing waves in a rotating column of fluid and constant magnetic field is examined. These standing waves are exact solutions to the nonlinear equations, and it is shown that these waves are unstable with respect to high-frequency, short wavelength perturbations. The overall growth rates of these instabilities are unaffected by the perturbations in the magnetic field in almost all but the most extreme parameter regimes. This perturbation analysis can be viewed as a secondary perturbation analysis of the rotating flow.

Accelerated diffusion in the centre of a vortex

The spiral wind-up and diffusive decay of a passive scalar in circular streamlines is considered. An accelerated diffusion mechanism operates to destroy scalar fluctuations on a time scale of order P1/3 times the turn-over time, where P is a Péclet number. The mechanism relies on differential rotation, that is, a non-zero gradient of angular velocity. However if the flow is smooth, the gradient of angular velocity necessarily vanishes at the centre of the streamlines, and the time scale becomes greater. The behaviour at the centre is analysed and it is found that scalar there is only destroyed on a time scale of order P1/2. Related results are obtained for magnetic field and for weak vorticity, a scalar coupled to the stream function of the flow. Some exact solutions are presented.

A SIMPLE DERIVATION OF THE CUTPOINT OF AN IMPACTOR

Impactors are designed according to numerical computations, which have been shown to give accurate predictions for the 50% cutpoint. However, such computer calculations obscure the basic physics of the impactor. Textbook derivations commonly assume circular streamlines and do not yield accurate results for the cutpoint. In this paper, a simple but accurate derivation is given for the cutpoint of an impactor with a rectangular nozzle, based on only very general assumptions as to the flow. For an impactor with a circular nozzle, the derivation requires an additional input datum and is less accurate. The derivation makes clear the essential physics of the impactor.

An analytic model for the prediction of incoherent shaped charge jets

It is now a well recognized fact that the jet from a shaped charge can be overdriven in the sense that the fastest moving particles are not produced as a cohesive mass of material. Rather, the tip material may be produced as a number of discrete particles which possess different nonzero radial components of velocity and hence spread out from the axis of symmetry of the charge. Such a jet is classed as incoherent and when this incoherency occurs the jet’s target penetration capability is invariably degraded. This physical phenomenon is the subject matter of this article. Several experimental results using common shaped charge materials are presented first. An analytic model which predicts the jet speed at the transition point between a coherent and incoherent state is then described. This model is based on the assumption that a stagnant core with circular boundaries exists in the flow region. Further, the flow field is assumed to be compressible with circular streamlines. The Murnaghan equation of state is used to relate the pressure and density in the flow region where the jet is produced. It is postulated that the transition between a coherent and incoherent state occurs when the circular flow becomes wholly supersonic. The critical Mach number for coherency is shown to be approximated to high accuracy by a simple formula depending on the collapse angle of the flow and the incoming flow speed. Excellent agreement between the model predictions and the experimental data is demonstrated.

Plane analog of spontaneous swirling

A plane analog of the problem of spontaneous swirling—the occurrence of a free transverse flow due to disturbance of the initial plane-parallel flow—is considered. It is shown that in flows with circular streamlines between coaxial cylinders, loss of stability can result in the occurrence of axial flow that is axisymmetric on the average (averaging over the axial coordinate and the azimuthal angle) because of the countergradient transfer of the axial momentum component by Reynolds stresses.

A new class of instabilities of rotating flows

The stability of flows which are the sum of a linear flow with circular or elliptic streamlines and a transverse standing wave is examined. A coordinate transformation annihilating the linear component of the flow is made, and the stability of the transformed flow is studied via two complementary methods. First, the stability with respect to local small scale perturbations is analyzed by virtue of the short wavelength stability method of Eckhoff and Lifschitz & Hameiri and it is found that all transformed flows are unstable with respect to such perturbations. Second, the corresponding linearized problem is studied via appropriately modified classical methods and global instabilities are found. It is shown that the growth rate of the global instabilities increases with their wavenumber and asymptotically approaches the value predicted by the short wavelength stability method. It is argued that the observed instabilities play an important role in transitions from laminar two-dimensional flows to turbulent three-dimensional ones.

A Galilean invariant explicit algebraic Reynolds stress model for turbulent curved flows

A Galilean invariant weak-equilibrium turbulence hypothesis that is sensitive to streamline curvature is proposed. The hypothesis leads to a fully explicit algebraic expression for Reynolds stress in terms of the mean velocity field and kinetic energy and dissipation of turbulence. The model is tested in curved homogeneous shear flow which is a homogeneous idealization of the circular streamline flow. The agreement is excellent with Reynolds stress closure model and adequate with available experimental data.

Circular streamline model of shaped-charge jet and slug formation with asymmetry

The formation of a jet as a result of the collapse of a shaped charge liner in the presence of asymmetry is considered. The development of a satisfactory analytic formation model based on the assumption of incompressible fluid flow requires the solution of the classically indeterminate problem of the collision of two unequal streams. A method of closing the problem is presented. It rests on the assumptions that there is a stagnant core region and that the flows of material from the impinging streams into the jet and slug turn by following circular streamlines with no decrease in speed. Balances of the centrifugal forces with the pressure in the stagnant core, relations derived from the flow geometry, the equation of mass conservation, and Bernoulli’s law provide the mathematical statement of the problem. These equations are manipulated to produce a reduced set of four equations in four unknowns, enabling a solution to be determined. This analytic solution predicts that both the jet and the slug are deflected by the same acute angle from the line of bisection of the angle between the impinging streams. The percentages of material in each stream which turn to form the jet are the same. The new model recovers analytically the classical Birkhoff, MacDougall, Pugh, and Taylor [J. Appl. Phys. 19, 563 (1948)] jet formation model in the symmetric case. It also yields the correct analytic result for the head-on collision of two streams of equal speeds but differing widths. More generally the model predicts an approximately linear dependence of the off-axis jet velocity component on the percentage difference in the stream speeds and a similar dependence for the widths. The predicted absolute values of the off-axis velocity are greater for a given difference in the stream speeds than for the same percentage difference in the widths. Finally, fair agreement with some previous experimental work on the collision of streams of unequal widths is demonstrated.

Normal forms for three-dimensional parametric instabilities in ideal hydrodynamics

We derive and analyze several low dimensional Hamiltonian normal forms describing system symmetry breaking in ideal hydrodynamics. The equations depend on two parameters (ϵ, λ), where ϵ is the strength of a system symmetry breaking perturbation and λ is a detuning parameter. In many cases the resulting equations are completely integrable and have an interesting Hamiltonian structure. Our work is motivated by three-dimensional instabilities of rotating columnar fluid flows with circular streamlines (such as the Burger vortex) subjected to precession, elliptical distortion or off-center displacement.

The instability of precessing flow

Abstract An explanation is put forward for the instability observed within a precessing, rotating spheroidal container. The constant vorticity solution for the flow suggested by Poincare is found to be inertially unstable through the parametric coupling of two inertial waves by the underlying constant strain field. Such resonant couplings are due either to the elliptical or shearing strains present which elliptically distort the circular streamlines and shear their centres respectively. For the precessing Earth's outer core, the shearing of the streamlines and the ensuing shearing instability are the dominant features. The instability of some exact, linear solutions for finite precessional rates is established and used to corroborate the asymptotic analysis. A complementary unbounded analysis of a precessing, rotating fluid is also presented and used to deduce a likely upperbound on the growth rate of a small disturbance. Connection is made with past experimental studies.

The instability of rotating fluid columns subjected to a weak external Coriolis force

Instabilities in rotating fluid columns subjected to a weak external Coriolis force are studied. External Coriolis force alters the base flow distorting circular streamlines of the unperturbed columns. The inviscid part of the modified flow (0,r,−2εr sin φ) is an exact solution of Euler equations. Here ε is the strength (nondimensional) of imposed Coriolis force. It is shown that this distortion leads to three-dimensional instabilities. The instability mechanism is generic. It occurs in many flows having circular streamlines distorted by a spectrum of modes cos mφ, sin mφ (m=2 corresponds to elliptical instability). The instabilities occur for wavelengths and frequencies at the intersection points of dispersion curves for the unperturbed columns. Moreover, the instabilities occur at the points of intersection for which modes are coupled with the external Coriolis mode. Here a case is studied where axisymmetric and helical modes are involved in interactions leading to fully three-dimensional flows. It is shown that rotating fluid columns are unstable to disturbances whose axial wavelengths lie in a band, whose width is proportional to the strength ε of imposed Coriolis force. Parametric resonance between a pair of inertial waves (natural modes of oscillation) and the external Coriolis mode is the physical explanation for the instabilities. Numerical results are presented on growth rates of mixed modes and on widths of unstable regions. The growth rates depend linearly on the strength of external Coriolis force. It is shown that viscosity shifts the instability tongues to positive values of ‖ε‖. The results of small-amplitude perturbation analysis are compared with full numerical simulations of the Navier–Stokes equations. Comments are made on competition between centrifugal and parametric instabilities in Taylor–Couette systems subjected to an external Coriolis force studied in Wiener et al. [J. Stat. Phys. 64, 913 (1991)] and Ning et al. [J. Stat. Phys. 64, 927 (1991)].

Nonlinear effects of non-newtonian fluids in unsteady flow with circular streamlines

The problem of the nonlinear effects of power law fluids in unsteady flow with circular streamlines is addressed. The governing equations are derived. The similarity solutions in closed form of these equations are obtained. The existence of traveling wave solutions for the distributions of tangential velocity and vorticity is shown. The cases of constant and variable angular velocity of the inner circular cylinder are considered and discussed. The presence of a moving front is found, which determines a finite velocity of disturbance propagation.

Nonaxisymmetric thermal equilibria of a cylindrically bounded guiding-center plasma or discrete vortex system

The thermal equilibria of a two-dimensional guiding-center model for a single-species plasma bounded by a cylindrical conductor are considered in the microcanonical ensemble. The same description applies to identical point vortices in a two-dimensional, ideal fluid surrounded by a circular streamline. The statistically dominant configurations are displaced asymmetrically from the axis, for sufficiently large energies at specified canonical angular momentum. The transition between symmetric and asymmetric states resembles a second-order phase transition, and occurs at negative temperatures. It is related to a bifurcation in the mean-field (Vlasov) description. The theory is compared with Monte Carlo simulations of microcanonical ensembles of guiding centers.

Tripolar vortices in a rotating fluid

THE emergence of coherent flow structures is a well-known feature of quasi-geostrophic or two-dimensional turbulence, and because of their relevance to large-scale geophysical flows the dynamics of these structures have been studied increasingly over the past decade1–5 The most common coherent structures are the axisymmetric (monopolar) vortex, with circular streamlines, and the vortex dipole, both of which have been found to arise in a variety of situations under different (sometimes nondescript) forcing conditions. The emergence of vortex dipoles, for example, has been observed in turbulence in soap films3, in electrically forced magnetohydrodynamic flows5 and in collapsing turbulence in a continuously stratified fluid6. There is growing evidence4,7–9 that, in addition to the monopolar and dipolar vortices, a tripolar coherent flow structure exists. Here we describe a laboratory experiment in which a tripolar structure is found to emerge from an unstable cyclonic vortex in a homogeneous rotating fluid. The tripole seems to be a very stable structure, even persisting in a highly sheared fluid environment, and might therefore be found in natural geophysical flows.

Non-Circular Gas Motions in the Inner Galaxy

It is shown that the observed motion of neutral hydrogen in the inner 1000 pc of the Galaxy is, for the most part, consistent with flow on circular streamlines in the potential of the Galactic bulge as derived from the observed distribution of near infrared emission. The implied mass distribution is also consistent with recent kinematic determinations of the stellar mass in the inner few parsecs of the bulge. The non-circular gas motion seen between two and four kpc is most likely due to flow on elliptical streamlines in the presence of a weak bar distortion of the Galactic disk. Circular gas motion in the region of the bulge and elliptical streaming further out is an observed characteristic of flow in barred galaxies and is consistent with our present theoretical understanding of such systems. The implication is that non-circular motions of the molecular clouds in the inner 200 pc have a non-gravitational origin. A possible mechanism for exciting such motions is an accretion event resulting from an encounter of a molecular cloud with a massive black hole. A starburst leading to a high supernovae rate 107 years ago in the inner 50 pc is an alternative explanation. Observations of molecular cloud regions in the nuclei of external normal galaxies could distinguish between alternative mechanisms.