Magnetic field effects on Newtonian and non-Newtonian ferrofluid flow past a circular cylinder

The changes in the flow properties under the action of electromagnetic body forces are investigated numerically for ferrofluid flow past a circular cylinder. Ferrofluid is modeled as both a Newtonian and a non-Newtonian Power-Law fluid. Magnetic forces are applied by placing magnets at different locations on the surface of the cylinder. The magnetostatic effects on the structure of the wake region, on drag reduction and on vortex formation length and frequency are shown and compared in terms of Reynolds number, interaction parameter, Power-Law index and magnet location. It is shown that the increase in the interaction parameter reduces drag for both Newtonian and non-Newtonian model. This decrease is observed to be higher for shear thinning and lower for shear thickening fluid compared to Newtonian case. It is also shown that vortex street formation in the wake region behind the cylinder may be delayed under high magnetic effects. The Strouhal number is higher for shear thinning case at both low and high Reynolds numbers, and lower for shear thickening case at high Reynolds numbers, compared to Newtonian fluid. The vortex formation frequency also decreases under the action of the magnetic field in all cases, however the vortex formation length increases. Placing the magnet towards the front region of the cylinder increases considerably the drag coefficient for both Newtonian and non-Newtonian model. This increase in drag coefficient is higher in the shear thinning fluid and lower in the shear thickening fluid compared to the Newtonian case.     

Influence of transfer,release, and absorption of energy on vortex properties of ideal fluids

The vortex properties of ideal fluid flows are examined. It is shown that the barotropy assumption in classical theorems of the conservation of vortex properties of a fluid can be replaced by the adiabaticity assumption. In that case, the connection between the formation and evolution of anomalous atmospheric phenomena and significant violations of adiabaticity in the movement of air masses can be traced.     

The critical cross-section of a vortex

Summary This paper discusses the problem of critical-flow cross-sections in vortex flows. It is shown that there are two different types of vortex flows, A-type and B-type vortices (say). An A-type vortex approaches its critical flow state as its cross-sectional area increases and departs from the critical state as the cross-sectional area is decreased. This property is associated with the particular dependence of total pressure and circulation on the stream function, and it holds for both subcritical and supercritical A-type vortices. On the other hand, both subcritical and supercritical B-type vortices approach their critical flow states as their cross-sectional areas are decreased and depart from their critical states for increasing cross-sectional area. As was shown by Benjamin, setting the first variation of the flow force with respect to the stream function equal to zero leads to Euler's equation of motion. The second variation also vanishes if the corresponding flow state is critical. In this case the sign of the third variation decides whether the flow is an A-type or a B-type vortex. Within the framework of inviscid-fluid flow theory the type of a vortex is preserved unless vortex breakdown occurs. Making use of the knowledge that vortex flows are controlled by two different types of critical-flow cross-sections a variety of vortex flow phenomena are investigated, including the two types of inlet vortices that are observed upstream of jet engines, the behavior of vortex valves, the flow characteristics of liquid-fuel atomizers and the bath tub vortex.     

Acoustic Radiation from a Circular Cylinder in a Subsonic Stream

The problem of acoustic radiation from a semi-infinite circularcylinder is discussed when a plane wave is propagated out ofthe mouth of the cylinder at which a vortex layer is attached.The effects of convection are included at low Mach numbers anda comparision is made of the radiation in the far field forthe case of propagation in (a) still air, (b) moving air withouta vortex layer and (c) moving air with a vortex layer. It isshown that in general the magnitude of the sound is much greaterdownstream than it is upstream. This directionality becomesmore pronounced in case (b) than in case (a) whilst in the presenceof a vortex layer the field is strengthened in the immediateregion downstream and weakened elsewhere.     

Generation of large-scale vortex dislocations in a three-dimensional wake-type flow

Numerical study of three-dimensional evolution of wake-type flow and vortex dislocations is performed by using a compact finite diffenence-Fourier spectral method to solve 3-D incompressible Navier-Stokes equations. A local spanwise nonuniformity in momentum defect is imposed on the incoming wake-type flow. The present numerical results have shown that the flow instability leads to three-dimensional vortex streets, whose frequency, phase as well as the strength vary with the span caused by the local nonuniformity. The vortex dislocations are generated in the nonuniform region and the large-scale chain-like vortex linkage structures in the dislocations are shown. The generation and the characteristics of the vortex dislocations are described in detail.     

On the short-wave nature of Richtmyer–Meshkov instability

In the case of a variable period (wavelength) of a perturbed interface, the instability and stability of Richtmyer–Meshkov vortices in perfect gas and incompressible perfect fluid, respectively, are investigated numerically and analytically. Taking into account available experiments, the instability of the interface between the argon and xenon in the case of a relatively small period is modeled. An estimate of the magnitude of the critical period is given. The nonlinear (for arbitrary initial conditions) stability of the corresponding steady-state vortex flow of perfect fluid in a strip (vertical periodic channel) in the case of a fairly large period is shown.     

Near-field fluctuations and far-field noise of a three-element airfoil system by a discrete vortex method

A circulation-based discrete vortex method is used on a three-element airfoil system. Kutta conditions and Kelvin’s circulation theorem are additional conditions required for this method to determine the circulation distributions on each element and to determine vortex shedding. Discrete shed vortices are introduced near the four sharp edges to represent the sharp-edge vortex shedding caused by unsteady flow separation. The computational procedure warrants neutrally stable solutions of the self-sustained fluctuating flowfield that can provide broad-band spectral information for far-field noise predictions. The near-field vortex method directly calculates the parameters used in an asymptotic formula for far-field sound computation that attributes the noise sources to vortex interactions among the shed vortices and the surface circulations of the three-element airfoil system. The far-field noise characters are then analyzed and compared to the experimental data in the literature.     

Interaction between Kirchhoff vortices and point vortices in an ideal fluid

We consider the interaction of two vortex patches (elliptic Kirchhoff vortices) which move in an unbounded volume of an ideal incompressible fluid. A moment second-order model is used to describe the interaction. The case of integrability of a Kirchhoff vortex and a point vortex by the variable separation method is qualitatively analyzed. A new case of integrability of two Kirchhoff vortices is found. A reduced form of equations for two Kirchhoff vortices is proposed and used to analyze their regular and chaotic behavior.     

Existence of KAM tori in the phase-space of lattice vortex systems

Under very mild conditions on the circulations, and for arbitrary vortex configurations, the existence of quasi-periodic solutions for a lattice vortex model is shown.Control over the size of the perturbation in the KAM-theory is achieved by uniform scalings of the circulations, the vortex separations, and time. Thus, additional restrictions on the circulations and the ratios of vortex separations are not required; this makes the result physically meaningful.     

On the bidirectional vortex and other similarity solutions in spherical coordinates

The bidirectional vortex refers to the bipolar, coaxial swirling motion that can be triggered, for example, in cyclone separators and some liquid rocket engines with tangential aft-end injectors. In this study, we present an exact solution to describe the corresponding bulk motion in spherical coordinates. To do so, we examine both linear and nonlinear solutions of the momentum and vorticity transport equations in spherical coordinates. The assumption will be that of steady, incompressible, inviscid, rotational, and axisymmetric flow. We further relate the vorticity to some power of the stream function. At the outset, three possible types of similarity solutions are shown to fulfill the momentum equation. While the first type is incapable of satisfying the conditions for the bidirectional vortex, it can be used to accommodate other physical settings such as Hill’s vortex. This case is illustrated in the context of inviscid flow over a sphere. The second leads to a closed-form analytical expression that satisfies the boundary conditions for the bidirectional vortex in a straight cylinder. The third type is more general and provides multiple solutions. The spherical bidirectional vortex is derived using separation of variables and the method of variation of parameters. The three-pronged analysis presented here increases our repertoire of general mean flow solutions that rarely appear in spherical geometry. It is hoped that these special forms will permit extending the current approach to other complex fluid motions that are easier to capture using spherical coordinates.     

Relative equilibrium and collapse configurations of four point vortices

Relative equilibrium configurations of point vortices in the plane can be related to a system of polynomial equations in the vortex positions and circulations. For systems of four vortices the solution set to this system is proved to be finite, so long as a number of polynomial expressions in the vortex circulations are nonzero, and the number of relative equilibrium configurations is thereby shown to have an upper bound of 56. A sharper upper bound is found for the special case of vanishing total circulation. The polynomial system is simple enough to allow the complete set of relative equilibrium configurations to be found numerically when the circulations are chosen appropriately. Collapse configurations of four vortices are also considered; while finiteness is not proved, the approach provides an effective computational method that yields all configurations with a given ratio of velocity to position.      

Topology of vortices in neutron stars

The structure of vortex solutions in a rotating neutron star is discussed. It is shown that the presence of additional topological charge gives rise to a significant modification of the vortex lattice.     

On the Instability of Gortler Vortices to Nonlinear Travelling Waves

Author to whom correspondence should be addressed Recent theoretical work by Hall & Seddougui (1989) has shownthat strongly nonlinear high-wavenumber Görtler vorticesdeveloping within a boundary layer flow are susceptible to asecondary instability which takes the form of travelling wavesconfined to a thin region centred at the outer edge of the vortex.This work considered the case in which the secondary mode couldbe satisfactorily described by a linear stability theory, andin the current paper our objective is to extend this investigationof Hall & Seddougui (1989) into the nonlinear regime. Wefind that, at this stage, not only does the secondary mode becomenonlinear, but it also interacts with itself so as to modifythe governing equations for the primary Görtler vortex.In this case, then, the vortex and the travelling wave driveeach other, and indeed the whole flow structure is describedby an infinite set of coupled nonlinear differential equations.We undertake a Stuart-Watson type of weakly nonlinear analysisof these equations and conclude, in particular, that on thisbasis there exist stable flow configurations in which the travellingmode is of finite amplitude. Implications of our findings forpractical situations are discussed, and it is shown that thetheoretical conclusions drawn here are in good qualitative agreementwith available experimental observations.     

The influence of turbulence on a columnar vortex with axial flow

The interaction between a columnar vortex and external turbulence is investigated numerically. A q -vortex is immersed in an initially isotropic homogeneous turbulence field, which itself is produced numerically by a direct numerical simulation of decaying turbulence. The formation of turbulent eddies around the columnar vortex and the vortex-core deformations are studied in detail by visualizing the flow field. In the less-stable case with q = –1.5, small thin spiral structures are formed inside the vortex core. In the unstable case with q = –0.45, the linear unstable modes grow until the columnar vortex make one turn. Its growth rate agrees with that of the linear analysis of Mayer and Powell[1]. After two turns of the vortex, the secondary instability is excited, which causes collapse of the columnar q -vortex and the sudden appearance of many fine scale vortices. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Formation of Vortex Rings

The production of a vortex ring formed by using a piston to drive fluid through an orifice is considered. A cylindrical vortex sheet is supposed to be formed initially which rolls up into a vortex ring. Energy and momentum are conserved during rollup and determine the speed and size of the ring. It is shown that these quantities are independent of the vorticity distribution in the core of the ring. Reasonable agreement with experimental observations is found. A speculation is made about the criterion for the rings to be laminar or turbulent.     

Pinning and mode-locking of reaction fronts by vortices

We present results of experiments on the behavior of reaction fronts in the presence of vortex-dominated flows. The flow is either a single vortex or a chain of vortices in an annular configuration, and the reaction is the excitable Belousov–Zhabotinsky chemical reaction. If the vortex chain oscillates periodically in the lateral direction, the reaction front often mode-locks to the oscillations, propagating an integer number of wavelengths of the flow (two vortices) in an integer number of drive periods. In the presence of a uniform “wind”, the front often freezes, remaining pinned to the leading vortex and neither propagating forward against the wind nor being blown backward by it. Studies with an individual vortex verify the ability of a moving vortex to pin and drag a reaction front. We use this pinning behavior to explain the mode-locking for the oscillating case.     

Aerodynamic characteristics of propeller in non-axial flow

Flow on running propellers installed on aircraft in general case is not axial what results in off-axis forces and moments at the propeller. Lifting line method is applied for propeller blade model using set of discrete vortex filaments. Propeller wake defined by vortex sheet of discrete vortex filaments is modelled with application of free-wake method. Presented propeller model combines the blade and the wake model and with its low computational cost realization it is suitable for preliminary prediction of propeller contribution to the aircraft aerodynamic characteristics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians

Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the Grothendieck Quot Scheme. The latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by Bradlow. This gives an effective way of computing certain intersection numbers (known as ``Gromov invariants') on the space of holomorphic maps into Grassmannians. We carry out these computations in the case where the Riemann surface has genus one.