Vorticity Singularities Research Articles

Numerical study of non-uniqueness for 2D compressible isentropic Euler equations

In this paper, we numerically study a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the initial data have an algebraic singularity in vorticity at the origin. These are different from the multi-dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of initial value problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. The compressible Euler equations are solved using the positivity-preserving discontinuous Galerkin method.

Multipole Vortex Blobs (MVB): Symplectic Geometry and Dynamics

Vortex blob methods are typically characterized by a regularization length scale, below which the dynamics are trivial for isolated blobs. In this article, we observe that the dynamics need not be trivial if one is willing to consider distributional derivatives of Dirac delta functionals as valid vorticity distributions. More specifically, a new singular vortex theory is presented for regularized Euler fluid equations of ideal incompressible flow in the plane. We determine the conditions under which such regularized Euler fluid equations may admit vorticity singularities which are stronger than delta functions, e.g., derivatives of delta functions. We also describe the symplectic geometry associated with these augmented vortex structures, and we characterize the dynamics as Hamiltonian. Applications to the design of numerical methods similar to vortex blob methods are also discussed. Such findings illuminate the rich dynamics which occur below the regularization length scale and enlighten our perspective on the potential for regularized fluid models to capture multiscale phenomena.

An analysis of the spatial spectrum of Hermite-Gauss beams with embedded optical vortex

This work deals with a detailed theoretical study of Hermite-Gauss beams with embedded vortical phase passing through a lens. Special attention is given to the location of vortical phase singularities in the focal plane. Sufficient conditions for the intensity null in the focal plane centre are found. The theoretical analysis is confirmed by numerical calculations.

Argand-plane vorticity singularities in complex scalar optical fields: An experimental study using optical speckle

The Cornu spiral is, in essence, the image resulting from an Argand-plane map associated with monochromatic complex scalar plane waves diffracting from an infinite edge. Argand-plane maps can be useful in the analysis of more general optical fields. We experimentally study particular features of Argand-plane mappings known as "vorticity singularities" that are associated with mapping continuous single-valued complex scalar speckle fields to the Argand plane. Vorticity singularities possess a hierarchy of Argand-plane catastrophes including the fold, cusp and elliptic umbilic. We also confirm their connection to vortices in two-dimensional complex scalar waves. The study of vorticity singularities may also have implications for higher-dimensional fields such as coherence functions and multi-component fields such as vector and spinor fields.

Slow Burgers vortices in hot conformal fluids

The quintessential vortex solution in (3+1)-dimensional nonrelativistic, incompressible fluid mechanics is the Burgers vortex. We show that, in a finite domain, conformal fluids also admit hot vortex solutions with everywhere nonrelativistic speeds. These are identical to Burgers' solution, except that their radius is reduced by a factor of 2/sqrt(3). A rough calculation indicates that at RHIC these vortices are indeed smaller than the fireball itself during thermalization. Similarly to the Burgers vortex, these solutions manifest vortex stretching which avoids short distance singularities and so suggests that conformal fluid flows share the same nonsingularity as solutions of the Navier-Stokes equations. Naively generalizing this calculation to an arbitrary equation of state w, we observe that the Burgers vortex radius diverges as w crosses -1. While it has been argued that such a crossover leads to an instability in certain perfect fluids, the absence of Burgers vortices and therefore vortex stretching suggests that, in addition to the well-studied big rip singularities, viscous phantom fluids generically develop vorticity singularities.

Momentum and scalar transport at the turbulent/non-turbulent interface of a jet

Conditionally sampled measurements withparticle image velocimetry(PIV) of a turbulent round submerged liquid jet in a laboratory have been taken atRe= 2 × 103between 60 and 100 nozzle diameters from the nozzle in order to investigate the dynamics and transport processes at the continuous and well-defined bounding interface between the turbulent and non-turbulent regions of flow. The jet carries a fluorescent dye measured with planarlaser-induced fluorescence(LIF), and the surface discontinuity in the scalar concentration is identified as the fluctuating turbulent jet interface. Thence the mean outward ‘boundary entrainment’ velocity is derived and shown to be a constant fraction (about 0.07) of the the mean jet velocity on the centreline. Profiles of the conditional mean velocity, mean scalar and momentum flux show that at the interface there are clear discontinuities in the mean axial velocity and mean scalar and a tendency towards a singularity in mean vorticity. These actual or asymptotic discontinuities are consistent with the conditional mean momentum and scalar transport equations integrated across the interface. Measurements of the fluxes of turbulent kinetic energy and enstrophy are consistent with computations by Mathew & Basu (Phys. Fluids, vol. 14, 2002, pp. 2065–2072) in showing that for a jet flow (without forcing) the entrainment process is dominated by small-scale eddying at the highly sheared interface (‘nibbling’), with large-scale engulfing making a small (less than 10%) contribution consistent with concentration measurements showing that the interior of the jet is well mixed. (Turbulent jets differ greatly from the free shear layer in this respect.) To explain the difference between velocity and scalar profiles, their conditional mean gradients are defined in terms of a local eddy viscosity and eddy diffusivity and the momentum and scalar fluxes inside the interface. Since the eddy diffusivity is larger than the eddy viscosity, the scalar profile is flatter inside the interface so that the scalar discontinuity is relatively greater than the mean velocity discontinuity. Theoretical arguments, following Hunt, Eames & Westerweel (inProc. of the IUTAM Symp. on Computational Physics and New Perspectives in Turbulence, ed. Y. Kaneda, vol. 4, 2008, pp. 331–338, Springer), are proposed for how the vortex sheet develops, how the internal structure of the interface layer relates to the inhomogeneous rotational and irrotational motions on each side and why the dominant entrainment process of jets and wakes differs from that of free shear layers.

On the regularity of some thermocapillary convection models

The classical model of confined thermocapillary convection is analyzed. Its vorticity singularity, independent of the contact angle, leads to infinite pressure values at contact lines, forbidding any numerical use of the Laplace equation to calculate free surface shapes. Four models are explored to overcome this difficulty: an explicit polynomial filtering, a Navier slip at the solid boundaries, an interface viscosity model and the combination of slip and interface viscosity. Regular solutions are obtained with the first and last approaches. Only the last one is based on physical considerations and, by the introduction of physical length scales, avoids infinite pressure values at the contact line.

Stability study of the floating zone with respect to the Prandtl number value

Thermocapillary convection in a laterally heated liquid bridge is studied numerically, using a Chebyshev spectral method. The stability of the axisymmetric basic state, with respect to 3D perturbations, is characterized over a large range of Prandtl number values (Pr∊[10−3,102]), thanks to the choice of a sufficiently sharp regularizing function of the vorticity singularities. Criteria are established to ensure the compatibility of this function with mass conservation, and to choose a correct model of the lateral heat flux in order to reach a spectral convergence of the results. First 3D nonlinear spectral computations are presented.

Fluid velocity fields derived from vorticity singularities

The present article addresses the existence by construction of random velocity fields produced by prescribed flow singularities which are consistent with the equations governing the motion of a two-dimensional viscous incompressible liquid. By showing first that the vorticity can be arbitrary and consistent with the flow equations it is then possible to determine the velocity field explicitly in terms of the vorticity and its derivatives.

Sensitivity of the liquid bridge hydrodynamics to local capillary contributions

In the usual models of thermocapillary flows, a vorticity singularity occurs at the contact free surface–solid boundaries. The steady axisymmetric hydrodynamics of the side-heated liquid bridge of molten metal is addressed here for its sensitivity to the size δ of a length scale explicitly introduced to regularize the problem. By linear stability analysis of the flows, various stable steady states are predicted: The already known steady states which are reflection-symmetric about the mid-plane, but also others which do not possess this property. The thresholds in Ma of the associated bifurcations are strongly dependent on δ, and converge with δ→0 towards low values. Published data give these results some physical relevance.

Probability current tornado loops in three-dimensional scattering

We consider scattering of a three-dimensional particle on a finite family of δ potentials. For some parameter values the scattering wavefunctions exhibit nodal lines in the form of closed loops, which may touch but do not entangle. The corresponding probability current forms vortical singularities around these lines; if the scattered particle is charged, this gives rise to magnetic flux loops in the vicinity of the nodal lines. The conclusions extend to scattering on hard obstacles or smooth potentials.

Bipolar outflows from stars and galaxies as a tornado phenomenon

At an early stage in the lives of stars and galaxies when they are surrounded by discs, vorticity in the disc concentrates into a central vortex, thus converting a Keplerian velocity field u ϕ α r −1/2 into an irrotational velocity field u ϕ α r −1, which implies inward transfer of angular momentum. Centrifugal forces due to spin-up of the inner region and gravity dominant in the outer region then squeeze gas at intermediate layers, increasing pressure gradient in the axial direction sufficiently to drive a wide-angle low-velocity bipolar outflow from the disc. A logarithmic singularity of vorticity at the axis implies strong centrifugal forces which expand plasma to radius R where pressure gradient balances centrifugal force density of ions; the much weaker centrifugal force density of electrons cannot balance pressure gradient, so that electrons are driven inwards relative to ions until charge separation limits the relative displacement. Now the radial gradient of u ϕ causes ions to rotate at a different rate to electrons, generating an azimuthal current density j ϕ which is the source of an axial magnetic field B z in the core of the central vortex. Centrifuging carries lines of B to the core wall, where they are wound into helical force-free configuration with B α j. An annular channel of radius R and thickness α R into which parallel helical lines of j and B are compressed constitutes a “magnetic vortex tube” (MVT). An MVT separates an inner high-velocity highly collimated outflow from the outer low-velocity wide-angle outflow, and is responsible for jets. Magnetic pinches in the MVT may constrict the core flow at HH objects.

A new scheme for vorticity computations near a sharp corner

The numerical method for treating the vorticity singularity of incompressible viscous flow around a re-entrant sharp corner is discussed. Based on comparisons of the ad hoc methods and the Moffatt expansion method, a new scheme, the vorticity-circulation method, is developed for contracting flow which is characterized by local flow acceleration and separation. The scheme is applied to the flow in a sudden contraction for a range of Reynolds numbers. The computational results are compared with published experimental data and other numerical methods.

Finite-time vortex singularity in a model of three-dimensional Euler flows.

An analytical model of three-dimensional Euler flows which exhibits a finite-time singularity is given. The singularity in vorticity occurs at a velocity field null (stagnation point) which lies on the line joining two vorticity field nulls. It is shown that the vorticity diverges inversely with time.

Incompressible potential flow about complete aircraft configurations

SummaryAn economic numerical method has been developed to calculate the incompressible potential flow about complete aircraft configurations by taking advantage of the computational efficiency of the internal distribution of singularities. For wing and wing-like components, source and vorticity singularities are distributed on the respective mean camber surfaces, while the fuselage carries a source distribution on its wetted surface. The singularity strengths are obtained by satisfying the flow tangency condition at selected points on all the wetted surfaces of the configuration. No attempt has been made to consider wing or body vortex shedding or separation. The results of the present method were compared with other theoretical and experimental data.

LOPAN - A low-order panel method for subsonic and supersonic flows

AbstractA low-order panel method is presented for the calculation of subsonic or supersonic linear flow about general configurations. The method uses piecewise constant source, doublet and/or vorticity singularities. The internal Dirichlet boundary condition is applied, providing zero perturbation potential inside the configuration. The external Neumann boundary condition may refer to a condition on the outward conormal component of the perturbation velocity (the so-called 'linearised normal mass flux'). However, the program may, if required, proceed to obtain specified conditions on the outward normal component of the perturbation velocity, using an iterative procedure.

Nonlinear growth of Kelvin–Helmholtz instability: Effect of surface tension and density ratio

The nonlinear evolution of initially small disturbances at an interface separating two fluids of different density and velocity, including surface tension effects, is investigated with the use of the vortex-sheet discretization approach. The location of the interface is tracked in time by following the motion of each vortex under the combined influence of all other vortices. The influence of surface tension and density discontinuity is incorporated in an equation governing the evolution of the circulation of each vortex. Increasing the surface tension or the density ratio is shown to reduce the growth of the disturbance. For density ratios larger than 0.2 a critical wavenumber exists that divides the unstable part of the spectrum into a region where a vorticity singularity can develop (with interface rollup) and a region where two finite vortical centers are formed (with partial or no rollup). For lower density ratios this bifurcation phenomenon is not observed.

Part II: Steady aerofoil-spoiler characteristics

SummaryA surface singularity method has been formulated to predict two-dimensional spoiler characteristics at low speeds. Vorticity singularities are placed on the aerofoil surface, on the spoiler surface, on the upper separation streamline from the spoiler tip and on the lower separation streamline from the aerofoil trailing edge. The separation region is closed downstream by two discrete vortices. The flow inside the separation region is assumed to have uniform total head.The downstream extent of the separated wake is an empirical input. The flows both external and internal to the separated regions are solved.Theoretical results have been obtained for a range of spoiler-aerofoil configurations which compare reasonably with experimental results.The model is deficient in that it predicts a higher compression ahead of the spoiler than obtained in practice. Furthermore, there is a minimum spoiler angle below which a solution is not possible; it is thought that this feature is related to the physical observation that at small spoiler angles, the separated flow from the spoiler reattaches on the aerofoil upper surface ahead of the trailing edge.

Calculation of Potential Flow about Arbitrary Three Dimensional Wings using Internal Singularity Distributions

SummaryA first order panel method has been developed for calculating the incompressible potential flow about arbitrary three-dimensional wings. The method utilises a distribution of source and vorticity singularities on the mean camber surface of the wing and solves for the distribution by satisfying the boundary condition of zero normal flow at selected points on the surface of the wing. The method takes less computing time compared to other existing first order methods for the comparable numerical accuracy. This method can handle wings having cusped trailing edges and thin sections.

Vortical singularity of self-similar gas flow in the reflection of a shock wave

Vortical singularity of self-similar gas flow in the reflection of a shock wave