Vorticity Method Research Articles

Desingularization of 2D elliptic free-boundary problem with non-autonomous nonlinearity

In this paper, we consider the existence and limiting behaviour of solutions to a semilinear elliptic equation arising from confined plasma problem in dimension two \[ \begin{cases} -\Delta u=\lambda k(x)f(u) & \text{in}\ D,\\ u= c & \displaystyle\text{on}\ \partial D,\\ \displaystyle - \int_{\partial D} \frac{\partial u}{\partial \nu}\,{\rm d}s=I, \end{cases} \] where $D\subseteq \mathbb {R}^2$ is a smooth bounded domain, $\nu$ is the outward unit normal to the boundary $\partial D$ , $\lambda$ and $I$ are given constants and $c$ is an unknown constant. Under some assumptions on $f$ and $k$ , we prove that there exists a family of solutions concentrating near strict local minimum points of $\Gamma (x)=({1}/{2})h(x,\,x)- ({1}/{8\pi })\ln k(x)$ as $\lambda \to +\infty$ . Here $h(x,\,x)$ is the Robin function of $-\Delta$ in $D$ . The prescribed functions $f$ and $k$ can be very general. The result is proved by regarding $k$ as a $measure$ and using the vorticity method, that is, solving a maximization problem for vorticity and analysing the asymptotic behaviour of maximizers. Existence of solutions concentrating near several points is also obtained.

A comparison of variational upwinding schemes for geophysical fluids, and their application to potential enstrophy conserving discretisations

Methods for stabilising the turbulent cascade of potential enstrophy are analysed and compared for a compatible finite element discretisation of the rotating shallow water equations. These different approaches to upwinding the potential vorticity include the well-known anticipated potential vorticity method (APVM), streamwise upwind Petrov-Galerkin (SUPG) method, and a more recent method where the trial functions are evaluated downstream within the reference element. In all cases the upwinding scheme conserves both potential vorticity and energy, since the antisymmetric structure of the equations is preserved. The APVM leads to a symmetric definite correction to the potential enstrophy that is dissipative and inconsistent. The SUPG scheme introduces a consistent correction to the APVM scheme that acts as a backscatter term ensuring a richer depiction of turbulent dynamics. The downwinded trial function formulation results in the advection of downwind corrections, which analysis and numerical experiments show to be quantitatively similar to the SUPG scheme for a turbulent shear flow. The main difference between the SUPG and downwinded trial function schemes is in the energy conservation and residual errors. If just two nonlinear iterations are applied then the energy conservation errors are improved for the downwinded trial function formulation, reflecting a smaller residual error than for the SUPG scheme.We also present new temporal formulations by which potential enstrophy is exactly integrated across each time level. Results using these formulations are observed to be stable in the absence of any dissipation, despite the aliasing of grid scale turbulence. Using such a formulation and the APVM with a coefficient O(100) times smaller that its regular value leads to turbulent spectra that are greatly improved at the grid scale over the SUPG and downwinded trial function formulations with unstable potential enstrophy errors.

Deformation conditions and kinematic vorticity within the footwall shear zone of the Wildhorse detachment system, Pioneer metamorphic core complex, Idaho

Detailed analysis of numerous sequential fabrics and microstructures preserved in the footwall shear zones of detachment fault systems can be used to clarify the spatial relationships and timing of deformation related to emplacement and exhumation of metamorphic core complexes. To quantify the kinematics during deformation, we conducted microstructural analyses along two ∼300 m transects across mylonitic sections of the footwall of the Eocene Wildhorse Detachment system within the Pioneer metamorphic core complex. These results enable comparison of deformation at different structural levels during exhumation and between monomineralic and polymineralic lithologies.In both transects, mylonitic rocks preserve normal-sense asymmetric structures such as recrystallized quartz oblique foliation, S/C–C′ structures, and asymmetric feldspar porphyroclasts. Quartz microstructures, CPO patterns, and kinematic vorticity analyses consistently indicate simple shear-dominated deformation, which is predicted for late stage, shallow-level deformation within a rolling hinge detachment system exhuming a metamorphic core complex. Variations in c-axis patterns and active slip systems are associated with temperature, influenced by structural level, and grain boundary processes related to lithology. To determine the degree of non-coaxiality and minimize limitations and uncertainties with kinematic vorticity methods, we used two methods to evaluate flow vorticity. Kinematic vorticity ranges from 0.55 to 0.95 for the oblique quartz foliation (δ/β) and C′-type shear bands (C′-c) methods. Quartz recrystallized grain size ranges from 10 to 37 μm, indicating maximum differential stresses of 80–110 MPa, which is consistent with the footwall shear zones of numerous metamorphic core complex detachment systems.

Analysis of the vortical flow in a cyclone using four vortex identification methods

This paper analyzes the vortex flow field and performance of a solid-gas cyclone separator using computational fluid dynamics (CFD) by employing the Reynolds stress turbulence model (RSM) and Eulerian-Lagrangian particle tracking scheme for particle diameter (d) range of 0.1 to 8 μm. The complex structure of vortices is analyzed using the vorticity method, λ2-criterion, Q-criterion, and Ω-method. The results demonstrate that the presence of helical guide vanes (HGVs) enhances the centrifugal force applied to the solid particles and fluid velocity up to 4.4 times compared to the conventional cyclone. It is revealed that the λ2, Q, and Ω methods provide more details of eddy structure in the inner and outer layers of the isovortex surfaces than the vorticity method. Besides, the vortex core diameter can be estimated by all vortex identification methods.

Influence of Rotating Speeds and Thermodynamic Effects on LOX Turbopump Performance

As a key factor of a reusable turbopump, axial thrust directly determines the reliability of the bears and seals, especially considering the influence of rotational speeds and thermodynamic effects. In this study, numerical simulations and experimental tests were conducted to explore the impact of rotational speeds and the divergences of the turbopump performance with/without the thermodynamic effects. The experimental data obtained at 15000 r/min are in good agreement with the numerical results, illustrating the validity of the numerical model. The results show that the head deviations based on the affinity laws are relatively smaller, only 0.1% difference at normal condition, yet the efficiency increases from 72.5% to 73.8%, which need to be modified through the formula correction. Notably, the axial thrusts acting on impeller are significantly affected by the rotational speeds, and the values at rated flowrate decrease 17.6% after increasing the operating condition to rated speed. The affinity laws are not applicable for obtaining the turbopump axial thrusts under different speeds. Furthermore, due to the changes of the local physical properties, the values of head coefficient and efficiency decrease 2.8% and 1.3% at rated flowrate, respectively, when the thermodynamic effects of liquid oxygen are considered. Meanwhile, the relative variations of axial thrust are between 1.46% and 3.74% within the whole flow range. Finally, an in-depth analysis of the internal flow was conducted through the velocity field and vorticity method. In conclusion, both the rotational speeds and thermodynamic effects significantly affect the performance of a cryogenic turbopump, especially the axial thrust, and the unsteady results of the rotor-stator cavity leakage flow need to be analyzed in the future.

Helical symmetry vortices for 3D incompressible Euler equations

In this paper, we study the desingularization of vortices for the 3D incompressible Euler equations in an infinite pipe. We construct a family of traveling-rotating helical vortices for the Euler equations with a general vorticity function, which tends asymptotically to singular helical vortex filament evolved by the binormal curvature flow. The results are obtained by using an improved vorticity method. Some asymptotic properties of this family of solutions have also been studied.

Spatiotemporal Prediction Method for Vorticity of Three-dimensional Near-wall Cylindrical Flow Based on Convolutional Recurrent Neural Network

Due to the existence of a wall surface, the flow around the near-wall cylinder is a very complex unsteady dynamic behavior. However, this kind of feature extraction and fast prediction method of complex flow is difficult to solve in engineering. Therefore, a data-driven model based on Convolutional Recurrent Neural Network (CRNN) is proposed to solve the feature extraction and fast prediction of complex flow, and to predict the vorticity of the near wake region of a three-dimensional near-wall cylinder with G/D=1. CRNN model can analyze the relationship between Spatiotemporal evolution. The three-dimensional flow around the near-wall cylinder with Reynolds number 1500 meshed, and the vorticity was calculated after grid calculation to establish the Spatiotemporal evolution data set. For machine learning problems with high dimensions and large amounts of data, the Adaptive Moment (Adam) estimation method served as an effectively optimize. According to the Spatiotemporal prediction results of vorticity, the prediction results of the model fit well with the numerical simulation results, and the vorticity of the future time step is successfully predicted on the space and time series, that is, the evolutionary relationship of the Spatiotemporal series of vorticity near the wake of the near-wall cylinder is established. After the above model is established, only the initial multi-time vorticity data need to be obtained. The model can serve as predicting the vorticity field in the near wake region of the near-wall cylinder in the future.

Analysis of the characteristics of the spectral orthogonal decomposed flow fields: Numerical and experimental investigations of the air flow field around a simplified container ship model

The instability of the wind resistance on container ships with a certain windward area often depends on the shedding of vortices on the surface of the hull. This affects the navigation of the ship. However, the flow field around the container ship is composed of lots of modes of different frequencies and energy, only some of them have close relationship with energy saving. In order to investigate the characteristics of the flow motions necessary, three-dimensional flow field around the container ship is simulated by Large Eddy Simulation(LES) and validated by model test. Then the two-dimensional simplified container ship model is adopted to highlight the characteristics of the flow field, the Reynolds number of the flow field analyzed at this time is 2.53×106. The flow field around the two-dimensional model is simulated and decomposed by spectral proper orthogonal decomposition (SPOD), it is found that the Strouhal numbers of vortex shedding obtained by λ2 criterion and vorticity method is consistent with that calculated by SPOD. The energy of the flow motion related to the present study is dominated by the primary flow structure decomposed by SPOD, and the high frequency motion obtained by SPOD has little contribution on it. An in-depth analysis of the flow field is performed in the present investigation by taking the flow field around a container ship for example, the further study of the three-dimensional problem is also expected.

Asymptotic behaviour of global vortex rings

In this paper, we are concerned with nonlinear desingularization of steady vortex rings in with a general nonlinearity f. Using the improved vorticity method, we construct a family of steady vortex rings which constitute a desingularization of the classical circular vortex filament in the whole space. The requirements on f are very general, and it may not satisfy the Ambrosetti–Rabinowitz condition. Some qualitative and asymptotic properties are also established.

From the flow to the polarization field: A cognitive way for ferroelectric vortex structures

The generation of topological domain structures can significantly influence the behaviors of ferroelectric materials, and, however, strategies and perspectives for treating and analyzing them are still lacking. Inspired by the fluid lines and vorticity in hydromechanics, a characterization method of polarization lines and polarization vorticity is proposed herein for characterizing the vortex structure evolution. The polarization vorticity has the same unit as charge density, and the polarization vortex size is identified as a structure for vorticity concentration. Observing the radius and vorticity of vortices, we find that the domain evolution can be divided into three stages and the vortices react to mechanical loading in two ways: increasing the vorticity and enlarging the radius. Moreover, vortices exhibit three-dimensional shape, just like the vortex tube, and their conservation is proved. This paper provides a basis for further research on topological domain structures and the mechanism of vortex evolution, which might help improve high energy density devices.

Vortex characteristics of a gas cyclone determined with different vortex identification methods

This paper characterizes the vortex flow field in a gas cyclone based on different vortex identification methods, including vorticity method, Q criterion, λ2 criterion, Ω method and Liutex method. Analysis and comparison of isosurfaces of different vortex identification methods indicate that except vorticity method, the vortex core, the eddy flow and the short-circuit flow can be identified directly by other methods. But the small scale vortices, such as top ash ring, can only be fully recognized by Liutex method. The contours demonstrate that the magnitude of five vortex parameters is the largest in the vortex core, followed by the short-circuit flow, and the eddy flow is the smallest. But only Liutex method can predict the vortex core center and identify the most severe vortex eccentricity is in the bottom of vortex finder, the lower part of the cone and the hopper. The work not only improves the understanding of the vortex structure in a gas cyclone, which can critically affects cyclone performance, but also reveals the differences in the commonly used vortex characterization methods.

Numerical investigation on the interaction between cavitation and vortex in a Venturi tube based on Liutex method

Abstract In this paper, the cavitating flow in a venturi tube is simulated by LES combined with the ZGB cavitation model. A satisfying agreement between the numerical and experimental results is obtained. The comparison among Liutex method, vorticity method, Q method, λ ci method and Ω method has shown that Liutex method could suppress interference of shearing and preserve intensity of vortex. Thus, Liutex vortex identification method was selected to analyse the interaction between cavitation and vortex based on our numerical results. It is found that the strong vortex ring in the throat induces the cavitation ring. The ring shape cavitation cloud develops into the diffuser and separates from the wall, where small vortices occur. Forced by the re-entrant flow and small vortices, the thickness of cavities reduces, resulting in the breaking and shedding of cavities. Shedding cavities rapidly collapse, strengthening the intensity of vortices around the cavities.

Traveling vortex pairs for 2D incompressible Euler equations

In this paper, we study desingularization of vortices for the two-dimensional incompressible Euler equations in the full plane. We construct a family of traveling vortex pairs for the Euler equations with a general vorticity function, which constitutes a desingularization of a pair of point vortices with equal intensities but opposite signs. The results are obtained by using an improved vorticity method.

Mixed convection in sinusoidal lid driven cavity with non-uniform temperature distribution on the wall utilizing nanofluid

Mixed convection heat transfer of Cu-water nanofluid in an arc cavity with non-uniform heating has been numerically studied. The top flat moving wall is isothermally cooled at Tc and moved with a constant velocity. While the heated arc stationary wall of the cavity is maintained at a hot temperature Th. FORTRAN code is used to solve the mass, momentum, and energy equations in dimensionless form with suitable boundary conditions. In this study, the Reynolds number changed from 1 to 2000, and the Rayleigh number changed from 0 to 107. Also, the range of nanoparticles volume fraction extends from ϕ = 0 to 0.07. Stream vorticity method selected for the discretization of flow and energy equations. The present results are compared with the previous results for the validation part, where the results found a good agreement with the others works. The isotherms are regulated near the arc-shape wall causing a steep temperature gradient at these regions and the local and average heat transfer rate increases with increased volume fraction or Reynolds number or Rayleigh number. Finally, Correlation equations of the average Nusselt number from numerical results are presented.

Flow past a rotating cylinder predicted by a compact Eulerian viscous vorticity method under non-inertial rotating frame

An Eulerian vorticity method, based on a vorticity-stream function formulation, initially developed for the 2-D viscous unsteady flow past a circular cylinder, is extended in the case of 2-D flow past a rotating cylinder in a rotational frame of reference. The vorticity equation is shown to have the same form as in the case of an inertial frame of reference. Thus, the general structure of the numerical algorithm remains the same as in the inertial frame of reference. The non-inertial effects only enter in the calculation of the velocity field, and the vorticity induced velocity is modified to account for the non-inertial contributions due to the frame rotation. Results from the current method for the flow past a rotating cylinder are presented and shown to agree very well with those from other numerical methods and experimental benchmark tests.

Study on characteristics of vortex structures and irreversible losses in the centrifugal pump

The development of flow control methods in the centrifugal pump relies on further understanding of the characteristics of vortex structures and their irreversible losses. In this paper, the detached eddy simulation is conducted on a model centrifugal pump, which shows a good agreement with the experiment. Combining with three generations of vortex identification methods (vorticity, Q criterion, omega and Liutex methods) and entropy production analysis, the results show omega and Liutex methods are highly recommended to analyze the vortex structures. Vorticity is the key factor to promote the energy dissipation, and the irreversible losses of vortex areas can be lower than their surroundings when there exists smaller vorticity or higher rotation strength. Concerning the pressure pulsation induced by vortex shedding, it is unreasonable to analyze this unsteady process via vortex structures identified by iso-surface because of the restriction of the threshold. In comparison, the change of integral length is more related to the pressure pulsation.

LES investigation on vortex dynamics of transient sheet/cloud cavitating flow using different vortex identification methods

The sheet/cloud cavitating flow always contains complex multiscale vortex structures generated by the cavity cloud shedding and collapsing. In this study, the transient sheet/cloud cavitating flow around a Clark-Y hydrofoil is numerically investigated using the Large Eddy Simulation (LES) method coupled with the Zwart–Gerber–Belamri (ZGB) cavitation model. The simulation accurately reproduces the unsteady cavitation evolution process, and the predicted time-averaged lift and drag coefficients, total vapor volume variation and velocity distribution agree fairly well with the experimental measurements. The cavitation vortex dynamics are studied in detail with different vortex identification methods including the vorticity method, the [Formula: see text]-criterion method, the [Formula: see text] method, the [Formula: see text] method and the Liutex method. The vortex identification ability of the different methods in the transient sheet/cloud cavitating flow is also discussed. Generally, the Liutex method combines the advantages of the other methods and can accurately identify both the vortex position and strength. Further analysis of cavitation-vortex interactions demonstrates that the cavity cloud shedding and collapsing have a pronounced influence on the vortex structure.

Vortex structure identification for supersonic flows

This work is devoted to the application of vortex structure identification methods for two counter-rotating wingtip vortices in a supersonic flow. The λ2 criterion and the method of maximum vorticity were used for detection vortex core lines and visualization of vortex sheet and rope. Computational model based on the URANS equations with SA turbulence model was applied. Numerical simulations were performed on the hybrid supercomputing system K-60 at the Keldysh Institute of Applied Mathematics RAS using the developed software package ARES for 3D turbulent flows modeling on high performance computing systems.

Desingularization of vortex rings in 3 dimensional Euler flows

In this paper, we are concerned with nonlinear desingularization of steady vortex rings of three-dimensional incompressible Euler fluids. We focus on the case when the vorticity function has a simple discontinuity, which corresponding to a jump in vorticity at the boundary of the cross-section of the vortex ring. Using the vorticity method, we construct a family of steady vortex rings which constitute a desingularization of the classical circular vortex filament in several kinds of domains. The precise localization of the asymptotic singular vortex filament is proved to depend on the circulation and the velocity at far fields of the vortex ring, and the geometry of the domains. Some qualitative and asymptotic properties are also established.

Kinematics and Timing Constraints in a Transpressive Tectonic Regime: The Example of the Posada-Asinara Shear Zone (NE Sardinia, Italy)

Detailed geological field mapping, integrated with meso- and microstructural investigations, kinematic of the flow and finite strain analyses, combined with geochronology, are fundamental tools to obtain information on the temperature–deformation–timing path of crystalline rocks and shear zone. The Posada-Asinara shear zone (PASZ) in northern Sardinia (Italy) is a steeply dipping km-thick transpressive shear zone. In the study area, located in the Baronie region (NE Sardinia), the presence of mylonites within the PASZ, affecting high- and medium-grade metamorphic rocks, provides an opportunity to quantify finite strain and kinematic vorticity. The main structures of the study area are controlled by a D2 deformation phase, linked to the PASZ activity, in which the strain is partitioned into folds and shear zone domains. Applying two independent vorticity methods, we detected an important variation in the percentage of pure shear and simple shear along the deformation gradient, that increases from south to north. We constrained, for the first time in this sector, the timing of the transpressive deformation by U–(Th)–Pb analysis on monazite. Results indicate that the shear zone has been active at ~325–300 Ma in a transpressive setting, in agreement with the ages of the other dextral transpressive shear zones in the southern Variscan belt.