Identification of vortex cores of three-dimensional large-vortical structures

In this paper, we propose a nonlocal method to identify vortex cores in three-dimensional flows as a complement to the existing list of local and nonlocal methods of the bibliography. The method is based on the vector field of the instantaneous rotation of a particle around a center. This center is defined using the Darboux vector field along the path-particle lines; the vortex core is detected using their Frenet–Serret frame. We illustrate the application of the method to identify the core of large-vortical structures in analytical and numerically simulated laminar and turbulent natural convection flows.     

Numerical simulations of internal solitary waves with vortex cores

This paper deals with the numerical verification of the theory developed by Derzho and Grimshaw (DG) (1997, Phys. Fluids 9(11), 3378–3385) regarding solitary waves in stratified fluids with recirculation regions. The Boussinesq approximation is made and the stratification is chosen such that the Brunt-Väisälä frequency differs only slightly from uniform stratification. To establish the consistency of the numerical scheme the usual KdV and mKdV solutions are tested first and then the solutions obtained by DG are considered. It is found that these waves remain of permanent form and are stationary when viewed at their corresponding phase speed. The recirculation region remains stagnant to first order as predicted by DG.     

Properties of similarity solutions for flows in turbulent vortex cores

A class of similarity solutions of the equations for turbulent vortex cores matching an external inviscid similarity flow with a power law of circumferential velocity variationv-r ^−m near the rotation axis and constant Bernoulli function is considered. Solutions are found to exist only in a certain range of the indexm of the exponential. For each suchm there are two solutions. The authors wish to apologise for a mistake which resulted in the figures in this paper corresponding to [1] and those in [1] corresponding to this paper. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 60–64, May–June, 1998. The work was financially supported by the Russian Foundation for Basic Research (project No. 95-01-00483).     

Certain features of similarity solutions for flows in viscous vortex cores

A class of steady similarity solutions of the equations for viscous vortex cores which correspond to external inviscid similarity solutions with a power-law variation of the circumferential velocityv-r ^−m near the rotation axis is considered. It is found that if the Bernoulli function in external flow is constant, then these solutions will exist only on a certain range of the indexm of the exponential. For eachm on this range there are two solutions. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 38–43, January–February, 1998. The work was financially supported by the Russian Foundation for Fundamental Research (project No. 95-01-00483).     

Waves on viscoelastic films

The stability of films of a viscoelastic fluid on an inclined plane is considered. The stress contains a time-dependent memory term and takes into account the surface tension effect. It is shown analytically and numerically that these flows can be stable or unstable depending on the Reynolds number. Profiles of the free surface are obtained as functions of the Reynolds numbers.Y. A. Berezin deceased 11.04.2004     

Experiments on jet/vortex interaction

An experimental study was performed to evaluate the effect of a cold jet on a single trailing vortex. Flow visualization and particle image velocimetry (PIV) measurements were conducted in wind and water tunnels. The main parameters were the ratio of jet-to-vortex strength, the jet-to-vortex distance, the jet inclination angle and the Reynolds number. It was shown that the jet turbulence is wrapped around the vortex and ingested into it. This takes place faster with decreasing jet-to-vortex distance and increasing jet strength. Both time-averaged and instantaneous flow fields showed that the trailing vortex became diffused with its rotational velocity and vorticity levels reduced when the jet is located close to the vortex. The mechanism with which the jet interacts with the vortex is a combination of vortices shed by the jet and the turbulence. No noticeable differences were found within the Reynolds number range tested. The effect of jet on the vortex is delayed when the jet is blowing at an angle to the free stream and away from the vortex such as during take-off.     

赫姆霍兹对流体涡旋运动的开创性研究

着重讨论赫姆霍兹在特定的历史背景下,最先对流体涡旋运动进行研究并创建了流体涡旋理论,他所给出的涡线、涡丝等概念至今仍在流体力学中沿用,以他的名字命名的赫姆霍兹定理和赫姆霍兹速度分解定理更是流体涡旋理论中重要的基本定理,现代流体涡旋理论正是在他的研究基础上发展起来的,并进一步指出：赫姆霍兹的理论无论在物理上还是数学上都具有重要意义,而且对开尔等人后来的研究工作产生了重要影响。     

Dynamics of vortex line in presence of stationary vortex

The motion of a thin vortex with infinitesimally small vorticity in the velocity field created by a steady straight vortex is studied. The motion is governed by non-integrable PDE generalizing the Nonlinear Schrodinger equation (NLSE). Situation is essentially different in a co-rotating case, which is analog of the defocusing NLSE and a counter-rotating case, which can be compared with the focusing NLSE. The governing equation has special solutions shaped as rotating helixes. In the counter-rotating case all helixes are unstable, while in the co-rotating case they could be both stable and unstable. Growth of instability of counter-rotating helix ends up with formation of singularity and merging of vortices. The process of merging goes in a self-similar regime. The basic equation has a rich family of solitonic solutions. Analytic calculations are supported by numerical experiment.     

Numerical simulation of secondary vortex chamber effect on the cooling capacity enhancement of vortex tube

A vortex tube with additional chamber is investigated by computational fluid mechanics techniques to realize the effects of additional chamber in Ranque–Hilsch vortex tube and to understand optimal length for placing the second chamber in order to have maximum cooling effect. Results show that by increasing the distance between two chambers, both minimum cold and maximum hot temperatures increase and maximum cooling effect occurs at Z/L = 0.047 (dimensionless distance).     

Transition process from one celled vortex to two celled vortex

Changes in the flow structure, the vorticity distribution and the stream function during the transition process from the one celled vortex to the two celled vortex following a rapid change in the radial Reynolds number have been examined. Starting from a thin filament-like structure corresponding to the one celled vortex. a thick vortex core first grew up from the outlet region and turned into a cylindrical structure corresponding to the two celled vortex. The vorticity distribution also changed from a single-summit type to a ring-summit type.     

Dispersion Equation for Water Waves with Vorticity and Stokes Waves on Flows with Counter-Currents

The two-dimensional free-boundary problem of steady periodic waves with vorticity is considered for water of finite depth. We investigate how flows with small-amplitude Stokes waves on the free surface bifurcate from a horizontal parallel shear flow in which counter-currents may be present. Two bifurcation mechanisms are described: one for waves with fixed Bernoulli’s constant, and the other for waves with fixed wavelength. In both cases the corresponding dispersion equations serve for defining wavelengths from which Stokes waves bifurcate. Necessary and sufficient conditions for the existence of roots of these equations are obtained. Two particular vorticity distributions are considered in order to illustrate the general results.     

Characterising shortwave instabilities on a vortex dipole

The paper presents a method for characterising shortwave instability on a vortex dipole. Such instabilities cause the initially straight vortex lines to deform in a sinuous manner. In order to quantify the phenomenon, it is necessary to (a) characterise the vortex dipole and (b) characterise instabilities developed on it. In this study, the vortex dipole characteristics were quantified by using a nonlinear least squares fit to a Lamb–Oseen vortex profile with the velocity field measured by means of particle image velocimetry. The instability was recorded by capturing images of hydrogen bubbles, which were used to mark the vortex centre line, with a CCD camera. It was characterised by applying a fast Fourier transform and seeking dominant wavenumber components and a representative amplitude based on a range of wavenumbers within a bandwidth of interest. The method was tested on simulated and real data. Using the simulated data, the shortwave instability growth rate was calculated with an uncertainty of better than 1% and the mean wavelength deduced with an uncertainty of 5.3%. Using real data, a constant initial growth rate was deduced in agreement with the established theory. Further work might improve the algorithms so that spatial variations in wavelength and growth rate can be determined.     

A vortex axis and vortex core border grid adaptation algorithm

In the design process of hydrodynamical and aerodynamical technical applications, the numerical simulation of massively separated vortical flow is crucial for predicting, for example, lift or drag. To obtain reliable numerical results, it is mandatory to accurately predict the physical behavior of vortices. Thus, the dominant vortical flow structures have to be resolved in detail, which requires a local grid refinement and certain adaptation techniques. In this paper, a vortex flow structure adaptation algorithm is presented, which is particularly designed for local grid refinement at vortex axes positions and associated vortex core border locations. To this end, a fast and efficient vortex axis detection scheme is introduced and the algorithm for the vortex core border determination is explained. As the interaction between vortices makes the assignment of grid points to a certain vortex axis difficult, a helicity‐based vortex distinction approach in combination with a geometrical rotational sensor is developed. After describing the combined different techniques in detail, the vortex feature adaptation algorithm is applied to analytical and more realistic examples, which show that the described grid adaptation algorithm is able to enhance the grid cell resolution locally such that all significant vortical flow phenomena are resolved. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical study of vortex reconnection for two anti-parallel vortex tubes

By direct numerical simulation of the Navier-Stokes equations we investigate the reconnection of two antiparallel vortex tubes. A new type of perturbation of the initial vorticity field is given which is different from that presented in Refs. [8] and [9]. The formation and the evolution of the “curved vortex belts”, their mutual action with the “bridges” are found. These are important phenomena not studied by others. The project supported by the LNM of Institute of Mechanics, Academia Sinica and The National Natural Science Foundation of China     

Study on multiple ring-like vortex formation and small vortex generation in late flow transition on a flat plate

A high-order direct numerical simulation (DNS) of flow transition over a flat plate at a free stream Mach number 0.5 has been carried out. During the simulation, we cannot find, according to the classical theory of boundary layer transition, the “hairpin vortex breakdown to smaller structures” in the last stage of flow transition on a flat plate. However, we did discover the so-called spikes as a result of a multibridge or multiring formation. This indicated a large and stable vortex structure which can travel for a long distance. We believe that this is a result of ring heads that are located in an inviscid region. These heads of the “turbulence spot” never seem to break down and persist as a stable structure. In addition, we discovered that the U-shaped vortex is a part of an existing coherent structure instead of a newly generated one. The U-shaped vortex also provides an additional channel to transfer vorticity to the ring from the wall. During travel, the leading primary ring in the front of the spot is sloped and skewed, causing disappearance of the second sweep. As a consequence, no energy is brought down to the lower boundary layer near the vortex ring head of the spot. Thus, the small length scale structures become damped and the existing U-shaped vortex structure becomes distinguishable. So, the question is where do the turbulent small length scale vortices come from? We will address this with a new theory which states that all of small length scales (turbulence) are generated by high-shear (HS) layers rather than being produced by “vortex breakdown.” The new DNS shows that the HS layers are produced by strong positive spikes surrounded by low-speed fluids and by the interaction between the secondary and higher-level vortices and the wall surface especially near the ring neck. This multiple ring-like vortex generation also follows the first Helmholtz vortex conservation law. Furthermore, the Λ-shaped vortex is formed and rolling up, and the Λ-vortex is stretched and narrowed, and a new bridge is generated after the neck. The bridge will further become a second ring and so on. Besides the original vortex legs, there are also U-shaped vortex tubes. Finally, the multiple ring vortex structure is formed. From this process, no evidence is found to support that two consequent multiring circles are mixed to generate small vortices. The connection of downdraft/updraft motions is also studied.     

Self-similar vortex reconnection

As shown by Crow in 1970, the evolution of two almost parallel vortex filaments with opposite circulation exhibits a long-wave instability. Ultimately, the symmetric mode increases its amplitude reconnecting both filaments and ending into the formation of an almost periodic structure of vortex rings. This is a universal process, which appears in a wide range of scales: from the vortex trails behind an airplane to a microscopic scale of superfluids and Bose–Einstein condensates. In this paper, I will focus on the vortex reconnection for the latter case by employing Gross–Pitaevskii theory. Essentially, I focus on the well-known laws of interaction and motion of vortex filaments. By means of numerical simulations, as well as theoretically, I show that a self-similar finite-time dynamics manifests near the reconnection time. A self-similar profile is selected showing excellent agreement with numerical simulations.     

Spatial Simple Waves on a Shear Flow

The paper studies simple waves of the shallowwater equations describing threedimensional wave motions of a rotational liquid in a freeboundary layer. Simple wave equations are derived for the general case. The existence of unsteady or steady simple waves adjacent continuously to a given steady shear flow along a characteristic surface is proved. Exact solutions of the equations describing steady simple waves were found. These solutions can be treated as extension of Prandtl–Mayer waves for sheared flows. For shearless flows, a general solution of the system of equations describing unsteady spatial simple waves was found.