Academic literature on the topic 'Kirchhoff vortex'

This paper presents two methods for predicting the noise from helicopter rotors in forward flight. Aerodynamic and acoustic solutions in the near field are computed with a finite-difference solver for the Euler equations. Two different Kirchhoff acoustics methods are then used to propagate the acoustic signals to the far field in a computationally-efficient manner. One of the methods uses a Kirchhoff surface that rotates with the rotor blades. The other uses a nonrotating Kirchhoff surface. Results from both methods are compared to experimental data for both high-speed impulsive noise and blade-vortex interaction noise. Agreement between experimental data and computational results is excellent for both cases. The rotating and nonrotating Kirchhoff methods are also compared for accuracy and efficiency. Both offer high accuracy with reasonable computer resource requirements. The Kirchhoff integrations efficiently extend the near-field finite-difference results to predict the far field helicopter noise.

We consider the Ginzburg–Landau equation in dimension two. We introduce a key notion of the vortex (interaction) energy. It is defined by minimizing the renormalized Ginzburg–Landau (free) energy functional over functions with a given set of zeros of given local indices. We find the asymptotic behaviour of the vortex energy as the inter-vortex distances grow. The leading term of the asymptotic expansion is the vortex self-energy while the next term is the classical Kirchhoff–Onsager Hamiltonian. To derive this expansion we use several novel techniques.

Explicit formulae for the Kirchhoff–Routh path functions (or Hamiltonians) governing the motion of N -point vortices in multiply connected domains are derived when all circulations around the holes in the domain are zero. The method uses the Schottky–Klein prime function to find representations of the hydrodynamic Green's function in multiply connected circular domains. The Green's function is then used to construct the associated Kirchhoff–Routh path function. The path function in more general multiply connected domains then follows from a transformation property of the path function under conformal mapping of the canonical circular domains. Illustrative examples are presented for the case of single vortex motion in multiply connected domains.

Tornadoes have been shown to radiate infrasound to great distances, however convincing fundamental sound mechanisms are still absent. After using vortex sound theory to study sound generated by two numerical tornadoes, we found that there is a significant low-frequency signal between 0.1 Hz and 1 Hz. The sound is closely related to rotation of the non-axisymmetric vorticity field and its frequency depends on the rotational frequency. The non-axisymmetric vorticity field is represented by a Kirchhoff vortex-like flow in baseline tornado model and by multiple-vortex flow in eddy injection tornado model. Interestingly, there also exist high-frequency components in the later model which are hypothesized to originate from vortex merging process. Field detection data of tornado infrasound provides some support for the low-frequency sound.

A centrifugal compressor of a micro turbine generator system is investigated by the theoretical model and numerical analysis to explore the characteristics of the tip leakage vortex as the centrifugal compressor approaches stall. The numerical simulation results show the cross-sectional shape of the tip leakage vortex is elliptical, and its long and short axes are gradually stretched as the compressor approaches stall. Moreover, the vortex trajectory is inclined to the pressure side of the adjacent blade. In addition, the Kirchhoff elliptical vortex model is introduced to analyze the flow passage constriction effect, the passage vortex squeezing effect, and the leakage flow translation effect. Results show that there is no upper limit for the flow passage constriction effect on the tip leakage vortex. Furthermore, relative to the original vortex, the minimum constriction effect depends on the axis ratio of the elliptical tip leakage vortex. The passage vortex has an expansion effect on the tip leakage vortex rather than a squeezing effect, which is limited and also depends on the axis ratio of the ellipse. However, the effect magnitude of the leakage flow depends on the scales both of the long and short axes, which also have no upper limit.

The propagation and scattering of vortex light beams in complex media have significant implications in the fields of laser imaging, optical manipulation, and communication. This paper investigates the scattering characteristics of vortex light beams from a random rough surface. Firstly, a two-dimensional Gaussian rough surface is generated using the Monte Carlo method combined with the linear filtering method. Subsequently, the vortex beams are decomposed into the superposition of infinite plane waves, and the scattering of each plane wave from the rough surface is calculated using the Kirchhoff Approximation method. Numerical results of the angle distribution and spatial distribution of OAM scattering Laser Radar Cross Section (LRCS) are presented, varying with different surface roughness parameters for a rough aluminum surface and the beam’s parameters. The results demonstrate that the scattering of vortex beams is influenced by the beam’s parameters, such as Orbital Angular Momentum (OAM) mode number and elevation angle, which may bring new insights into vortex wave-matter interactions and their applications in high resolution imaging.

Cette étude est consacrée à la diffusion d'ondes acoustiques par des tourbillons isolés et couches de cisaillement de jet turbulentes. Lorsque les ondes acoustiques traversent un volume de turbulence, les fluctuations de la turbulence modifient la direction de propagation des ondes. En outre, si la turbulence évolue dans le temps, le contenu spectral du son change également, ce qui entraîne un élargissement spectral. Afin de mieux comprendre ces phénomènes, une série d'analyses numériques a été réalisée. Pour ce faire, un code fourni par Siemens a été utilisé, dans lequel les Equations d'Euler Linéarisées sont résolues par la Méthode de Galerkin Discontinue. Il simule la propagation des ondes acoustiques sur un écoulement de base défini par l'utilisateur. Pour prendre en compte l'élargissement spectral, le code a été modifié pour pouvoir interpoler en temps et en espace des données externes dépendant du temps dans l’écoulement de base. L'interpolation a été testée par des différentes études de convergence du champ de pression diffusé par une couche de mélange bidimensionnelle. D'autres caractéristiques ont également été mises en œuvre pour faire face aux instabilités numériques causées par l'inhomogénéité de l’écoulement de base. Dans un premier temps, la diffusion des ondes acoustiques causée par un tourbillon elliptique de Kirchhoff isolé est étudiée. Lorsque le tourbillon est fixé dans l'espace, l'étude se concentre sur les effets de l'ellipticité, de l'orientation du tourbillon par rapport à la direction de propagation de l'onde acoustique incidente, de la vitesse tangentielle du tourbillon et de sa taille par rapport aux ondes acoustiques. La diffusion a été également étudiée lorsque le tourbillon est convecté. Une attention particulière a été accordée à son ellipticité et à la vitesse de convection. Les résultats montrent que l'ellipticité et surtout l'orientation du tourbillon jouent un rôle clé dans la diffusion. Enfin, l'étude de la diffusion du son par les couches de cisaillement des jets turbulents est menée, où la source acoustique est située à l'axe du jet. Pour cela, les données interpolées dans l'écoulement de base du code DGM appartiennent à une base de données externe de jets ronds simulés par LES. Ces jets ont des nombres de Mach variant entre 0,3 et 1,3, et leur température est 1, 1,5 ou 2,25 fois la température ambiante. Ces paramètres modifient les propriétés des fluctuations turbulentes. Le contenu spectral de ces fluctuations est donc comparé entre les jets. Ensuite, les champs de pression obtenus avec des écoulements de base moyens et des écoulements de base turbulents, ainsi que la différence entre eux, sont présentés. Leurs directivités sont également discutées, ainsi que les spectres du champ acoustique. Les spectres sont caractérisés par une tonalité centrale à la fréquence de la source et deux lobes latéraux. Ils sont symétriques pour des nombres de Mach élevés. La position des lobes latéraux se rapproche du ton central et leurs niveaux augmentent avec la température du jet pour des jets à nombre de Mach constant, ce qui peut s'expliquer par les changements subis par les fluctuations de la turbulence
This study is consecrated to the scattering of acoustic waves by isolated vortices and turbulent jet shear layers. When the acoustic waves pass through a volume of turbulence, the fluctuations in the turbulence change the propagation direction of the waves. In addition, if the turbulence evolves in time, there is also a change in the sound spectral content, causing spectral broadening. In order to better understand these phenomena, a series of numerical analyses have been carried out. For this purpose, a code provided by Siemens has been used where the Linearised Euler Equations are solved by the Discontinuous Galerkin method. It simulates the acoustic wave propagation over a base flow defined by the user. To take into account the spectral broadening, the code has been modified to be able to interpolate time-dependent external data in time and space onto the base flow. The interpolation has been tested by different convergence studies of the pressure field scattered by a 2-dimensional mixing layer. Other features have been also implemented to cope with the numerical instability waves caused by the inhomogeneity of the base flow. Initially, the scattering of acoustic waves caused by an isolated Kirchhoff elliptic vortex is investigated. When the vortex is fixed in space, the study focuses on the effects of the ellipticity, the orientation of the vortex regarding the direction of propagation of the incident acoustic wave, the tangential velocity of the vortex and its size regarding the acoustic waves. The scattering has been investigated also when the vortex is convected. Special attention has been devoted to its ellipticity and the velocity convection. The results show that the ellipticity and especially the orientation of the vortex play a key role in the scattering. Finally, the study of the scattering of sound by turbulent jet shear layers is conducted, where the acoustic source is located at the jet axis. For that, the data interpolated in the base flow of the DGM code belong to an external database of round jets simulated by LES. These jets have Mach numbers varying between 0.3 and 1.3, and their temperature is 1, 1.5 or 2.25 times the ambience temperature. These parameters modify the properties of the turbulent fluctuations. Therefore, the spectral content of these fluctuations is compared between the jets. After that, the pressure fields obtained with mean base flows and turbulent base flows, and the difference between them are presented. Their directivities are also discussed, as well as the spectra of the acoustic field. The spectra are characterized by a central tone at the source frequency and two lateral lobes. They are symmetric for high Mach numbers. The position of the lateral lobes shifts closer to the central tone and their levels increase with the jet temperature for jets with constant Mach number, which can be explained by the changes undergone by the turbulence fluctuations

Quasi-geostrophic dynamics being essentially the vortex dynamics, the main notions of vortex dynamics in the plane are introduced in this chapter. Dynamics of vorticity is treated both in Eulerian and Lagrangian descriptions. Dynamics of point vortices and vortex patches (contour dynamics) are recalled, as well as discretisations of the vorticity equation preserving Casimir invariants, which reflect Lagrangian conservation of vorticity. The influence of the beta effect upon vortices is illustrated, and exact modon solutions of the QG equations on the f and beta planes are constructed. Basic notions of turbulence and specific features of two dimensional turbulence are reviewed for future use. Lighthill radiation of gravity waves by vortices is illustrated on the example of a pair of point vortices, and back-reaction of the radiation upon the vortex system is demonstrated and analysed. Influence of rotation upon the Lighthill radiation is explained. Construction of the Kirchhoff vortex solution is proposed as a problem.

Abstract This initial chapter starts with an investigation and discussion of Hamiltonian systems and applications. The chapter then continues with a canonical formulation of classical mechanics. The chapter then examines and elaborates on the classical many-body problems. Next, the chapter focuses on Kepler’s three laws of planetary motion, which is followed by a discussion of Newton’s law of gravitation. The chapter also studies the Kirchhoff point vortices as well as the thermodynamics of a DNA double-helix chain. It also covers the topics of the motion of a massive particle, the Helmholtz-Kirchhoff vortex model, partition function, and the dynamic modeling of DNA denaturation

It has been identified that vorticity in a vortex core directly relates to the frequency of a significant sound peak from an aircraft wake vortex pair where each of the vortices is modeled as an elliptic core Kirchhoff vortex. In three-dimensional vortices, sinusoidal instabilities at various length scales result in significant flow structure changes in these vortices, and thus influence their radiated acoustic signals. In this study, a three-dimensional vortex particle method is used to simulate the incompressible vortical flow. The flow field, in the form of vorticity, is employed as the source in the far-field acoustic calculation using a vortex sound formula that enables computation of acoustic signals radiated from an approximated incompressible flow field. Cases of vortex rings and a pair of counter-rotating vortices are studied when they are undergoing both long- and short-wave instabilities. Both inviscid and viscous interactions are considered and effects of turbulence are simulated using sub-grid-scale models.

Uni-axially tensioned wide webs and narrow ribbons commonly used in the paper-handling, textile, sheet-metal, and plastics industries are known to undergo large amplitude vibrations characterized as aeroelastic flutter. The aeroelastic stability of stationary wide webs and narrow ribbons coupled with fluid flow across the free edges of the web or ribbon is investigated in this article. The web or ribbon is modeled as a uni-axially tensioned Kirchhoff plate with vanishingly small bending stiffness. The 3D unsteady fluid flow surrounding the web or ribbon is evaluated numerically by using the vortex-lattice method. Wide webs are mainly found to exhibit the divergence instability. For some values of the applied tension, the clustered web modes exhibit frequency curve veering accompanied by a weak flutter instability before the occurrence of the divergence instability. The applied tension plays a critical role in deciding the type of instability in narrow ribbons. In cross flow, depending on the applied tension, narrow ribbons undergo flutter instability or divergence instability or the simultaneous onset of both instabilities.

Numerical predictions of fan noise have not been studied extensively. This is due to the scattering effect of the fan casing, duct and the difficulty in obtaining aerodynamic acoustic source. New method to predict the fan noise and performance is developed and used to calculate various fan noise problems. A vortex method is used to model the fan and to calculate the flow field. Acoustic pressures are obtained from the unsteady force fluctuations of the blades using an acoustic analogy. But the acoustic analogy can be applied only in the free field in general. In order to consider the solid boundary effects of the casing, the newly developed Kirchhoff-Helmholtz BEM (Boundary Element Method) is introduced. With the above-mentioned method, the flow field and sound field of centrifugal and axial fan were calculated. Reasonable results are obtained not only for the peak frequencies but also for the amplitudes of the tonal sound. Also, in the predicted sound field, we can see the scattering effect of duct and casing.