Kirchhoff Vortex Research Articles

Euler fluid in two dimensions: Statistical approach.

We use Kirchhoff's vortex formulation of 2D Euler fluid equationsto explore the equilibrium state to which a 2D incompressible fluid relaxes from an arbitrary initial flow. The vortex dynamics obeys Hamilton's equationsof motion with x and y coordinates of the vortex position forming a conjugate pair. A state of fluid can, therefore, be expressed in terms of an infinite number of infinitesimal vortices. If the vortex dynamics is mixing, the final equilibrium state of the fluid should correspond to the maximum of Boltzmann entropy, with the constraint that all the Casimir invariants of the fluid must be preserved. This is the fundamental assumption of Lynden-Bell's theory of collisionless relaxation. In this paper, we will present a Monte Carlo method which allows us to find the maximum entropy state of the fluid starting from an arbitrary initial condition. We will then compare this prediction with the results of molecular dynamics simulation and demonstrate that the final state to which the fluid evolves is, actually, very different from that corresponding to the maximum of entropy. This indicates that the mixing assumption is not correct. We will then present a different approach based on core-halo distribution which allows us to accurately predict the final state to which the fluid will relax, starting from an arbitrary initial condition.

Вплив внутрішньої динаміки когерентних вихрових структур, що взаємодіють між собою, на генероване звукове поле

The article is dedicated to the glowing memory of the talented Ukrainian scientist-mechanic, professor, doctor of physical and mathematical sciences Vyacheslav Vladimirovich Meleshko. The sound fields of several coherent vortex structures interacting with each other were calculated using the moment model (MZS-model) of the first and second orders. In the first-order MZS-model, vortices are described by point vortices, in the second-order MZS-model--by Kirchhoff vortices. Sound fields as a result of vortex interaction are calculated using the Lighthill's acoustic analogy under Powell's formulation. The spectrum of the sound field in the description of coherent vortices by point vortices is characterized by a single frequency band, which reflects the motion of the vortex centers of vortices (large-scale vortex movements). Taking into account the inner vortex dynamics leads to the expansion of spectrum and level increasing of calculated sound: the lower band describes the motion of the vorticity centers of spots, the higher band describes the internal dynamics of the vorticity. It is shown that large-scale vortex movements can be modeled by point vortices, when the sound spectrum of distributed vortices is clearly divided into bands and the width of the lower frequency band is close to the width of the sound spectrum from point vortices. The presence of an upper frequency band in the sound field of interacting vortex spots does not allow us to describe the distributed vortices as point vortices when constructing a sound source, since such a simplification leads to a significant underestimation of the level of the calculated sound field.

Finite element generation of sibilants /s/ and /z/ using random distributions of Kirchhoff vortices

The numerical simulation of sibilant sounds in three-dimensional realistic vocal tracts constitutes a challenging problem because it involves a wide range of turbulent flow scales. Rotating eddies generate acoustic waves whose wavelengths are inversely proportional to the flow local Mach number. If that is low, very fine meshes are required to capture the flow dynamics. In standard hybrid computational aeroacoustics (CAA), where the incompressible Navier-Stokes equations are first solved to get a source term that is secondly input into an acoustic wave equation, this implies resorting to supercomputer facilities. As a consequence, only very short time intervals of the sibilant can be produced, which may be enough for its spectral characterization but insufficient to synthesize, for instance, an audio file from it or a syllable sound. In this work, we propose to substitute the aeroacoustic source term obtained from the computational fluid dynamics (CFD) in the first step of hybrid CAA, by a random distribution of Kirchhoff's spinning vortices, located in the region between the upper incisors and the lower lip. In this way, one only needs to solve a linear wave equation to generate a sibilant, and therefore avoids the costly large-scale computations. We show that our proposal can recover the outcomes of hybrid CAA simulations in average, and that it can be applied to generate sibilants /s/ and /z/. Modeling and implementation details of the Kirchhoff vortex distribution in a stabilized finite element code are discussed in the paper, as well as the outcomes of the simulations.

Inertia–gravity wave radiation from the elliptical vortex in the f-plane shallow water system

Inertia–gravity wave (IGW) radiation from the elliptical vortex is investigated in the f-plane shallow water system. The far field of IGW is analytically derived for the case of an almost circular Kirchhoff vortex with a small aspect ratio. Cyclone–anticyclone asymmetry appears at finite values of the Rossby number (Ro) caused by the source originating in the Coriolis acceleration. While the intensity of IGWs from the cyclone monotonically decreases as f increases, that from the anticyclone increases as f increases for relatively smaller f and has a local maximum at intermediate f. A numerical experiment is conducted on a model using a spectral method in an unbounded domain. The numerical results agree quite well with the analytical ones for elliptical vortices with small aspect ratios, implying that the derived analytical forms are useful for the verification of the numerical model. For elliptical vortices with larger aspect ratios, however, significant deviation from the analytical estimates appears. The intensity of IGWs radiated in the numerical simulation is larger than that estimated analytically. The reason is that the source of IGWs is amplified during the time evolution because the shape of the vortex changes from ideal ellipse to elongated with filaments. Nevertheless, cyclone–anticyclone asymmetry similar to the analytical estimate appears in all the range of aspect ratios, suggesting that this asymmetry is a robust feature.

Explicit Equations to Transform from Cartesian to Elliptic Coordinates

Explicit equations are obtained to convert Cartesian coordinates to elliptic coordinates, based on which a function in elliptic coordinates can be readily mapped in physical space. Application to Kirchhoff vortex shows that its elliptical core induces two counter-rotating irrotational eddies.

Bifurcation of rotating patches from Kirchhoff vortices

In this paper we investigate the existence of a new family of rotating patches for the planar Euler equations. We shall prove the existence of countable branches bifurcating from the ellipses at some implicit angular velocities. The proof uses bifurcation tools combined with the explicit parametrization of the ellipse through the exterior conformal mappings. The boundary is shown to belong to Holderian class.

Reynolds stress and the energy balance of a localized two-dimensional vortex in a uniform shear flow

Consideration is given to the kinetic energy balance of a localized two-dimensional vortex in unbounded space, subject to a uniform background shear flow. For this problem, a quadratic invariant based on the total flow can be constructed that consists of the sum of the vortex self-energy and the energy of interaction with the background flow. It is shown that an energy equation also may be written for the rate of change of vortex self-energy, relating this to the rate of working by the Reynolds stress. The stress integral is demonstrated to converge for a localized vortex of finite circulation, in contrast to the total kinetic energy. The two approaches to the energy balance are shown to be complementary, and the relation between the Reynolds stress and interaction energy is discussed. As an example, the integrated Reynolds stress is evaluated for a uniformly sheared elliptical (Kirchhoff) vortex. The stress integral includes far field contributions, indicating that appreciable exchange of energy with the external flow occurs well beyond the boundary of the vortex.

Reactive linearized equations of perturbed compressible variables for low-Mach number variable-density flows

A hydrodynamic/acoustic splitting approach is proposed to study noise emitted from reactive variable-density flows. A simulation using the variable-density low-Mach number equations provides a solution of the hydrodynamic motions of the flow (base-flow), and a set of equations for perturbed variables is additionally solved to capture the acoustic motions. A rigorous derivation of these equations for the assumed base-flow is given, which compared to its non-reacting counterpart includes additional terms related to variable-density flows. Two different test cases are presented. First, the Kirchhoff vortex is simulated to highlight instability issues related to constant-density flows. Second, an academic test case for variable-density flows is proposed in the form of a reacting dipole, which is used to underline the stability of the proposed perturbed equations in such a scenario. Various intermediate forms of the derived perturbation equations are juxtaposed and analyzed with respect to their stability for these two test cases, and assumptions made in their derivation are numerically justified.

The evolution of Kirchhoff elliptic vortices

A Kirchhoff elliptic vortex is a two-dimensional elliptical region of uniform vorticity embedded in an inviscid, incompressible, and irrotational fluid. By using analytic theory and contour dynamics simulations, we describe the evolution of perturbed Kirchhoff vortices by decomposing solutions into constituent linear eigenmodes. With small amplitude perturbations, we find excellent agreement between the short time dynamics and the predictions of linear analytic theory. Elliptical vortices must have aspect ratios less than a∕b=3 to be completely stable. At late times, unstable perturbations evolve to states consisting of filaments surrounding and connecting one or more separate vortex core regions. Even modes have two different evolution paths accessible to them, dependent on the initial phase. Ellipses can first fission into more than one separate region when a∕b=6.046, from the negative branch m=4 mode. Increasing the perturbation amplitude can result in nonlinear instability, while the perturbation is still small relative to any vortex dimension. For the lowest m modes, we quantify the transition from linear to nonlinear behavior.

Influence of Vortex Core on Wake Vortex Sound Emission

A possible sound emission mechanism of aircraft wake vortices has been identified by using both measurement data and theoretical results. This mechanism relates consistently to a dominant frequency of sound pressure matching the rotation frequency of a Kirchhoff vortex. The rotation of the Kirchhoff vortex is due to the self-induction inside the vortex core. Numerical simulations are necessary for a more realistic wake consisting of a counterrotating vortex pair with inviscid ground effects and shear flow. The simulations are carried out using a vortex particle method. A far-field vortex sound formulation, based on asymptotic expansions, is developed to calculate acoustic pressure from the wake vortices. The simulation results confirm that the frequency of the wake sound emission is essentially the same as the classical Kirchhoff vortex, even under the influences of an inviscid ground effect or a weak-shear cross wind. The aforementioned results suggest that the identified mechanism should be fairly robust, which demonstrates its persistency under several different environmental conditions.

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.

Linearized perturbed compressible equations for low Mach number aeroacoustics

For efficient aeroacoustic computation at low Mach numbers, the linearized perturbed compressible equations (LPCE) are proposed. The derivation is based on investigation of the perturbed vorticity transport equations. In the original hydrodynamic/acoustic splitting method, perturbed vorticity is generated by a coupling effect between the hydrodynamic vorticity and the perturbed velocities. At low Mach numbers, the effect of perturbed vorticity on sound generation is not significant. However, the perturbed vorticity easily becomes unstable, and causes inconsistent acoustic solutions, based on grid dependence. The present LPCE ensures grid-independent acoustic solutions by suppressing the generation of perturbed vorticity in the formulation. The present method is validated for various dipole and quadruple vortex-sound problems at low Mach numbers: (i) laminar dipole tone from a circular cylinder at Reynolds number based on the cylinder diameter, Re D = 150 and free stream Mach number, M ∞ = 0.1, (ii) quadruple sound of Kirchhoff vortex at Mach number based on the rotating speed, M Θ = 0.1, and (iii) temporal mixing layer noise at Reynolds number based on the shear layer thickness, Re δ = 10000 and Mach number based on the shear rate, M s = 0.1.

Взаимодействие вихрей Кирхгофа и точечных вихрей в идеальной жидкости

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.

Internal gravity wave emission from a pancake vortex: An example of wave–vortex interaction in strongly stratified flows

At small Froude numbers the motion of a stably stratified fluid consists of a quasisteady vortical component and a propagating wave component. The vortical component is organized into layers of horizontal motions with well-pronounced vertical vorticity and often takes the form of so-called “pancake” vortices. An analytical model of such a vortex that is a solution of the Euler–Boussinesq equations at a vanishing Froude number is constructed as a superposition of horizontal two-dimensional Kirchhoff elliptic vortices. This vortex is nonstationary and internal gravity waves are, therefore, excited by its motion. The radiation properties are studied by matching the vortex field with the far internal gravity wave field according to the procedure applied in acoustics to determine vortex sound. The structure of the gravity wave field is completely quantified. By calculating energy and angular momentum fluxes carried by outgoing waves and attributing them to the adiabatic change of the vortex parameters, we calculate the backreaction of the internal gravity waves radiation and show that, as in the case of acoustic radiation by the Kirchhoff vortex, this adiabatic evolution leads to an elongation of the vortex, and its eventual destabilization.

High order numerical simulation of sound generated by the Kirchhoff vortex

An improved high order finite difference method for low Mach number computational aeroacoustics (CAA) is described. The improvements involve the conditioning of the Euler equations to minimize numerical cancellation errors, and the use of a stable non-dissipative sixth-order central spatial interior scheme and a third-order boundary scheme. Spurious high frequency oscillations are damped by a third-order characteristic-based filter. The objective of this paper is to apply these improvements in the simulation of sound generated by the Kirchhoff vortex.

Entropy Splitting for High Order Numerical Simulation of Vortex Sound at Low Mach Numbers

Several recent developments in efficient, stable, highly parallelizable high order non-dissipative spatial schemes with characteristic based filters that exhibit low dissipation for long time linear and nonlinear wave propagations are utilized for computational aeroacoustics (CAA). For stability consideration, the Euler equations are split into a conservative and a symmetric non-conservative portion. Due to the large disparity of acoustic and stagnation quantities in low Mach number aeroacoustics, the split Euler equations are formulated in perturbation form to minimize numerical cancellation errors. Spurious oscillations are suppressed by a characteristic-based filter. The method has been applied to accurately simulate the sound emitted by an almost circular Kirchhoff vortex at low Mach numbers.

The strain rate in evolutions of (elliptical) vortices in inviscid two-dimensional flows

In this paper the strain rate in evolutions of elliptical vortices in inviscid two-dimensional flows is considered. Previous work [Vosbeek et al., Phys. Fluids 9, 3315 (1997)] has revealed that there exists a relationship between the (in)stability of an elliptical vortex of uniform vorticity (Kirchhoff vortex) and the spatial distribution of the strain rate. It is examined here how the strain distribution evolves in time for unstable Kirchhoff vortices. Furthermore, it is shown that there also exists a relationship between strain and (in)stability of a vortex consisting of nested elliptical patches representing a more general vorticity distribution.

Geometric phases for corotating elliptical vortex patches

We describe a geometric phase that arises when two elliptical vortex patches corotate. Using the Hamiltonian moment model of Melander, Zabusky, and Styczek [J. Fluid Mech. 167, 95–115 (1986)] we consider two corotating uniform elliptical patches evolving according to the second order truncated equations of the model. The phase is computed in the adiabatic setting of a slowly varying Hamiltonian as in the work of Hannay [J. Phys. A 18, 221–230 (1985)] and Berry [Proc. R. Soc. London, Ser. A 392, 45–57 (1984)]. We also discuss the geometry of the symplectic phase space of the model in the context of nonadiabatic phases. The adiabatic phase appears in the orientation angle of each patch—it is similiar in form and is calculated using a multiscale perturbation procedure as in the point vortex configuration of Newton [Physica D 79, 416–423 (1994)] and Shashikanth and Newton [J. Nonlinear Sci. 8, 183–214 (1998)], however, an extra factor due to the internal stucture of the patch is present. The final result depends on the initial orientation of the patches unlike the phases in the works of Hannay and Berry [J. Phys. A 18, 221–230 (1985)]; [Proc. R. Soc. London, Ser. A 392, 45–57 (1984)]. We then show that the adiabatic phase can be interpreted as the holonomy of a connection on the trivial principal fiber bundle π:T2×S1→S1, where T2 is identified with the product of the momentum level sets of two Kirchhoff vortex patches and S1 is diffeomorphic to the momentum level set of two point vortex motion. This two point vortex motion is the motion that the patch centroids approach in the adiabatic limit.

Three-dimensional instability of a multipolar vortex in a rotating flow

In this paper, the elliptic instability is generalized to account for Coriolis effects and higher order symmetries. We consider, in a frame rotating at the angular frequency Ω, a stationary vortex which is described near its center r=0 by the stream function written in polar coordinates Ψ=−(r2/2)+p(rn/n)cos(nθ), where the integer n is the order of the azimuthal symmetry, and p is a small positive parameter which measures the strength of the nonaxisymmetric field. Based on the Lifschitz and Hameiri [Phys. Fluids A 3, 2644–2651 (1991)] theory, the local stability analysis of the streamline Ψ=−1/2 is performed in the limit of small p. As for the elliptic instability [Bayly, Phys. Rev. Lett. 57, 2160–2163 (1986)], the instability is shown to be due to a parametric resonance of inertial waves when the inclination angle ξ of their wave vector with respect to the rotation axis takes a particular value given by cos ξ=±4/(n(1+Ω)). An explicit formula for the maximum growth rate of the inertial wave is obtained for arbitrary ξ, Ω, and n. As an immediate consequence, it is shown that a vortex core of relative vorticity Wr (assumed positive) is locally unstable if Ω<−(1+n/4)Wr/2 or Ω>(−1+n/4−p(n−1)/2)Wr/2. The predictive power of the local theory is demonstrated on several vortex examples by comparing the local stability predictions with global stability results. For both the Kirchhoff vortex and Moore and Saffman vortex, it is shown how global stability results can be derived from the local stability analysis using the dispersion relation of normal (Kelvin) modes. These results are compared to those obtained by global methods and a surprisingly good agreement is demonstrated. The local results are also applied to rotating Stuart vortices and compared to available numerical data.

A Possible Mechanism for the Eye Rotation of Typhoon Herb

An elliptical eye that rotated cyclonically with a period of approximately 144 minutes in Typhoon Herb 1996 was documented. The elliptical region had a semimajor axis of 30 km and a semiminor axis of 20 km. Two complete periods of approximately 144 min were observed in the Doppler radar data. The rotation of the elliptical eye in the context of barotropic dynamics at three levels were explored: linear waves on a Rankin vortex, a nonlinear Kirchhoff vortex, and with a nonlinear spectral model. The linear wave theory involves the existence of both the high (potential) vorticity gradient near the eye edge and the cyclonic mean tangential flow in the typhoon. The propagation of (potential) vorticity waves in the cyclonic mean flow makes the elliptical eye rotate cyclonically. The rotation period is longer than the period of a parcel trajectory moving in the cyclonic mean flow around the circumference, because the vorticity wave propagates upwind. The nonlinear theory stems from the rotation of Kirchhoff’s vortex. Estimates of the eye rotation period from both linear and nonlinear theories agree with observations of the eye rotation period when the observed maximum wind from Herb is used. Nonlinear numerical computations suggest the importance of the interaction of neutral vorticity waves, which determine the shape and the rotation period of the eye. The calculations also support the rotation of the eye in approximately 144 min in the presence of axisymmetrization, vorticity redistribution, wave breaking, and vortex merging processes.