Vortex Filaments Research Articles

Helical vortices with small cross-section for 3D incompressible Euler equation

In this article, we construct traveling-rotating helical vortices with small cross-section to the 3D incompressible Euler equations in an infinite pipe, which tend asymptotically to singular helical vortex filament evolved by the binormal curvature flow. The construction is based on studying a general semilinear elliptic problem in divergence form{−ε2div(K(x)∇u)=(u−q|ln⁡ε|)+p,x∈Ω,u=0,x∈∂Ω, for small values of ε. Helical vortex solutions concentrating near several helical filaments with polygonal symmetry are also constructed.

A different modelling of complex Hasimoto map for pseudo-null curves via Bishop frame

In this paper, we construct some links between visco model of motion and equations attained by taking geometric differential description of Hasimoto map of pseudo-null curves framed by the Bishop frame and some physical consequences. First, we find that the equations of Hasimoto transformation are occurred by adapting Hasimoto map for pseudo-null curves. Then, we investigate the correspondence of both the Landau–Lifshitz system and the vortex filament equation by means of the Hasimoto map of pseudo-null curves. Lastly, we portray the links of findings obtained by geometric physical issues. Finally, we simulate optical solutions with graphically physical constructions.

Hidden Degrees of Freedom in Implicit Vortex Filaments

This paper presents a new representation of curve dynamics, with applications to vortex filaments in fluid dynamics. Instead of representing these filaments with explicit curve geometry and Lagrangian equations of motion, we represent curves implicitly with a new co-dimensional 2 level set description. Our implicit representation admits several redundant mathematical degrees of freedom in both the configuration and the dynamics of the curves, which can be tailored specifically to improve numerical robustness, in contrast to naive approaches for implicit curve dynamics that suffer from overwhelming numerical stability problems. Furthermore, we note how these hidden degrees of freedom perfectly map to a Clebsch representation in fluid dynamics. Motivated by these observations, we introduce untwisted level set functions and non-swirling dynamics which successfully regularize sources of numerical instability, particularly in the twisting modes around curve filaments. A consequence is a novel simulation method which produces stable dynamics for large numbers of interacting vortex filaments and effortlessly handles topological changes and re-connection events.

Pseudo null growth model and its classifications based on generalized vortex filament equation

The pseudo null growth model in Minkowski 3-space is shaped by evolving a pseudo null curve as denoted direction, growth velocity, and stretch in this paper. Meanwhile, the idea of the generalized Hasimoto surfaces is proposed based on the vortex filament equation. The conditions and geometric structures of the generalized Hasimoto pseudo null growth surfaces are investigated accompanied by some typical examples.

Direct observation of a stable spiral wave reentry in ventricles of a whole human heart using optical mapping for voltage and calcium.

From the studies of Gordon Moe in the 1960s, functional reentry has been investigated as a mechanism for tachycardia and fibrillation; however, it was not until 1990 that optical mapping (OM) experiments using sheep hearts verified the existence of reentrant spiral waves.1 Eventually, it was shown that electromechanical 3-dimensional vortex filaments drive cardiac fibrillation.2 To date, only multiple complex and short-lived reentrant waves have been recorded using OM on human ventricles.3 The presence of single spiral waves in human ventricles has only been demonstrated using low-resolution intracardiac electrocardiograms (ECGs).

Duct sound produced by vortex flow over a splitter plate

This paper examines the sound generated by an inviscid low Mach number vortex filament as it moves near a semi-infinite splitter in a two-dimensional duct with and without vortex shedding at the splitter edge. A strong one-dimensional sound is produced when the vortex is close to the edge. Without vortex shedding, the sound is shown to consist of predominantly compressive pulses which propagate in the split duct section and a rarefaction pulse downstream of the splitter edge, regardless of the position of the half plate and the direction of vortex motion. The vortex shedding results in no horizontal force at and no sound going downstream of the edge. Results also suggest that the vortex shedding does not always result in noise reduction. In all cases with and without vortex shedding, the narrower section of the split duct experiences the most sound in general. The surprising result is found that the confinement of the edge-scattering flow within a duct does not much affect the high acoustic efficiency of the scattering process.

A Gradient Tensor–Based Subgrid-Scale Parameterization for Large-Eddy Simulations of Stratified Shear Layers Using the Weather Research and Forecasting Model

Abstract The transition process from laminar stratified shear layer to fully developed turbulence is usually captured using direct numerical simulations, in which the computational cost is extremely high and the numerical domain size is limited. In this work, we introduce a scale-aware subgrid-scale (SGS) parameterization, based on the gradient tensor of resolved variables, which is implemented in the Weather Research and Forecasting (WRF) Model. With this new SGS model, we can skillfully resolve the characteristics of transition process, including formation of vortex cores, merging vorticity billows, breaking waves into smaller scales, and developing secondary instability in the stratified shear layer even at coarse-resolution simulations. Our new model is developed such that the time scales of the eddy viscosity and diffusivity terms are represented using the tensor of the gradient and not that of the rate-of-strain, which is commonly used in the parameterization of turbulent-viscosity models. We show that time scales of unresolved transition processes in our new model are correlated with those of vorticity fields. At early times, the power-law slopes in the kinetic and available potential energy spectra are consistent with the process of formation and merging waves with an upscale energy transfer. At later times, the power-law slopes are in line with the process of breaking waves into small-scale motions with a downscale transfer. More importantly, the efficiency of turbulent mixing is mainly high at the edge of vortex filaments and not at the vortices’ eyes. These findings can improve our understanding of turbulent mixing process in large-scale wind-induced events, such as tropical cyclones, using the WRF Model. Significance Statement The evolution of instabilities in stratified shear layers has significant impacts on the structure of large-scale geophysical flows and also on the energy pathway to smaller-scale motions in internal waves and turbulence. Resolving transition processes in stratified shear layers requires very high-resolution simulations in climate models. We propose a new subgrid-scale parameterization that is implemented in the Weather Research and Forecasting Model to capture the dynamics of transition process from laminar to three-dimensional turbulence in stratified shear layers at coarse-resolution simulations. Our new scale-aware parameterization can reduce biases in climate models by skillfully representing unresolved fluxes, leading to higher accuracy in weather predictions of temperature, precipitation, and surface fluxes with an affordable computational cost.

Closely spaced co-rotating helical vortices: long-wave instability

We consider as base flow the stationary vortex filament solution obtained by Castillo-Castellanos et al. (Phys. Rev. Fluids, vol. 6, 2021, 114701) in the far wake of a rotor with tip-splitting blades. The cases of a single blade and of two blades with a hub vortex are studied. In these solutions, each blade generates two closely spaced co-rotating tip vortices that form a braided helical pattern in the far wake. The long-wave stability of these solutions is analysed using the same vortex filament framework. Both the linear spectrum and the linear impulse response are considered. We demonstrate the existence of different types of instability modes. A first type corresponds to the local pairing of consecutive turns of the helical pattern, which is well described by the instability of a uniform helical vortex with a core size given by the mean separation distance of the vortices in the pair. A second type corresponds to the pairing of consecutive turns of the vortex pair and is observed only for densely braided patterns, which is well described by the instability of two interlaced helical vortices by straightening out the baseline helix. A third type of unstable modes modifies the separation distance between the vortices in each pair and amplifies specific (linear) wavelengths. These unstable modes also spread spatially with a weaker rate.

Interaction of quasi-two-dimensional vortical gusts with airfoils, unswept and swept wings

A nearly two-dimensional vortex of small core size has been produced by the transient plunging motion of an upstream airfoil and it interacted with the downstream wings. Depending on the offset distance of the vortex and wing angle of attack, the incident vortex filament deforms, diffuses, and loses coherence, while inducing leading-edge vortex formation and shedding from the wing. No significant spanwise flow develops in the incident vortices during the interaction. The interaction with the swept wing at each spanwise plane appears to be unaffected by the other spanwise planes. The counter-clockwise vortex induces a positive lift peak as it approaches the wing, which can be predicted by the potential flow assumption. The peak lift force is proportional to the circulation of the incident vortex and has its maximum near the zero-offset distance. The minimum lift coefficient is reached after the vortex has just passed and caused flow separation on the lower surface. The maximum lift coefficients for the finite unswept and swept wings can be estimated by making a correction for the aspect ratio and using the independence principle. The only exception is observed for the swept wing at a post-stall angle of attack for which the leading-edge vortex shedding becomes parallel to the leading-edge and increases the peak lift force.Graphical abstract

Dynamics of tip vortices on plunging wings

The vortex dynamics of the tip vortices on oscillating wings in periodic and transient plunging motions are investigated via three-dimensional velocimetry measurements in a water tunnel. The ratio of the maximum plunging velocity to the freestream velocity Vpmax/U∞ is varied in the range of 0.1 to 1.5. At lower plunging velocity and frequency, the tip vortex appears quasi steady as a straight vortex filament. With increasing frequency and amplitude of the wing motion, three-dimensional deformations appear as helical waves on the vortex filament while merging with the trailing-edge vortices also takes place. The instability waves appear as the circulation of the tip vortex approaches near its maximum during the cycle. The wavelength normalized by the vortex core radius varies in the range of λ/a=2 to 4 for the periodic as well as transient plunging motion. We found reasonable agreement with the theoretical predictions for isolated vortices. It is likely that the presence of the wing flow as well as other vortices result in the strain and disturbances that excite the tip vortex core. We propose that the mechanism of generation of instabilities on tip vortices is similar to those of the leading-edge and trailing-edge vortices.

Interacting helical vortex filaments in the three-dimensional Ginzburg–Landau equation

For each given n\ge 2 , we construct a family of entire solutions u_\varepsilon (z,t) , \varepsilon > 0 , with helical symmetry to the three-dimensional complex-valued Ginzburg–Landau equation \Delta u+(1-|{u}|^2)u=0, \quad (z,t) \in \mathbb{R}^2\times \mathbb{R} \simeq \mathbb{R}^3. These solutions are 2\pi/\varepsilon -periodic in t and have n helix-vortex curves, with asymptotic behavior, as \varepsilon\to 0 , u_\varepsilon (z,t) \approx \mathop{\smash[t]{\prod_{j=1}^n}\vphantom{\prod}} W\bigl(z- \varepsilon^{-1} f_j(\varepsilon t)\bigr), where W(z) =w(r) e^{i\theta} , z= re^{i\theta} , is the standard degree +1 vortex solution of the planar Ginzburg–Landau equation \Delta W+(1-|{W}|^2)W=0 in \mathbb{R}^2 and f_j(t) = \frac{\sqrt{n-1} e^{it}e^{2 i (j-1)\pi/ n}}{\sqrt{|{\log\varepsilon}|}}, \quad j=1,\ldots, n. Existence of these solutions was previously conjectured by del Pino and Kowalczyk (2008), \mathbf{f}(t) = (f_1(t),\ldots, f_n(t)) being a rotating equilibrium point for the renormalized energy of vortex filaments derived there, \mathcal{W}_\varepsilon (\mathbf{f}) := \pi \int_0^{2\pi} \Biggl( \frac{|{\log\varepsilon}|} 2 \sum_{k=1}^n |{f'_k(t)}|^2 - \sum_{j\neq k}\log |{f_j(t)-f_k(t)}| \Biggr) \operatorname{d}t, corresponding to that of a planar logarithmic n -body problem. The modulus of these solutions converges to 1 as |{z}| goes to infinity uniformly in (t) , and the solutions have nontrivial dependence on t , thus negatively answering the Ginzburg–Landau analogue of the Gibbons conjecture for the Allen–Cahn equation, a question originally formulated by H. Brezis.

Covector fluids

The animation of delicate vortical structures of gas and liquids has been of great interest in computer graphics. However, common velocity-based fluid solvers can damp the vortical flow, while vorticity-based fluid solvers suffer from performance drawbacks. We propose a new velocity-based fluid solver derived from a reformulated Euler equation using covectors. Our method generates rich vortex dynamics by an advection process that respects the Kelvin circulation theorem. The numerical algorithm requires only a small local adjustment to existing advection-projection methods and can easily leverage recent advances therein. The resulting solver emulates a vortex method without the expensive conversion between vortical variables and velocities. We demonstrate that our method preserves vorticity in both vortex filament dynamics and turbulent flows significantly better than previous methods, while also improving preservation of energy.

Optical modeling for recursional Hasimoto map with normalized time fractional applications

In this paper, we construct the optical flow equations obtained with the help of a timelike curve in De Sitter 2D space S12 and with the aid of the Hasimoto map defined by this curve. First, we introduce the Hasimoto map, which we call the De-Hasimoto map, for a timelike curve in S12. Then, with the help of this map and the equation representing the vortex filament motion, we obtain some flow equations with time and arc length parameters. Finally, we compute exact solutions of time fraction of an obtained optical motion equation, and we present some physical reviews on it.

Simplified vortex methods to model wake vortex roll-up in real-time simulations for fuel-saving formation flight

One possibility for reducing fuel consumption is to fly in the upwind field of the wake vortex generated by an aircraft that is flying ahead. Migratory birds use this principle. Manually flying an aircraft at the point of optimal fuel reduction is not suited for routine flight operations as the pilot workload is excessively high. Hence, an autopilot function has to carry out this task. For designing the autopilot, a flight mechanical simulation with a wake vortex velocity model is required that has the ability to calculate the vortex-induced velocity fields. This paper contributes to the choice of a real-time simulation method for modelling vortex-induced velocities that the wake vortex of a leading aircraft generates and that the trailing aircraft shall use during fuel-saving formation flight.Two different wake vortex velocity models are introduced and compared during steady, horizontal flight. One model is based on the Lifting Line Method (LLM) and the other on the unsteady Vortex Lattice Method (VLM). Both models are able to calculate the wake vortex roll-up phase for arbitrary lift distributions, whereas the commonly used Single Horseshoe Vortex Model (SHVM) ignores the near-field roll up. The differences in the induced upwind distribution and vortex filament position are analysed for coarse spatial and temporal discretisation that the real-time constraint requires. Despite the more stringent simplifications of LLM, both methods yield similar filament positions and similar velocity fields for the same discretisation of the lifting surfaces. Finally, the influence of the discretisation parameters is discussed and parameter values are recommended for using VLM and LLM in real-time flight simulations.

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.

Contactless generation and trapping of hydrodynamic knots in sessile droplets by acoustic screw dislocations

Helicity is an important quantity in fluid mechanics that indicates the presence of linked or knotted hydrodynamic vortex filaments. Such flow structures are not only promising elementary structures to study mass and momentum transfer in turbulent flows but also potent analogs for other topological problems arising in particle physics, liquid crystals, and plasma physics. However, experimental studies of knots and links are highly challenging due to the limited control over helicity generation and difficult observation of the resulting fast-paced multiscale flow evolution. In this paper, we propose using acoustic streaming to link hydrodynamic filaments in fluids. The method is contactless, almost instantaneous, and relatively insensitive to viscosity. Importantly, it allows starting from quite arbitrary three-dimensional flow structures without relying on external boundary conditions. We demonstrate our approach by using an acoustic screw dislocation to link two hydrodynamic vortex filaments in a sessile droplet. We observe an inversion of the flow chirality (measured by the hydrodynamic helicity) as the topological charge of the screw dislocation is increased. Combined with recent progress in acoustic field synthesis, this work opens a window to study more complex hydrodynamic knots and links topology at a broader range of space and time scales.

An Analytical Study of Flatness and Intermittency through Riemann's Nondifferentiable Functions

In the study of turbulence, intermittency is a measure of how much Kolmogorov's theory of 1941 deviates from experiments. It is quantified with the flatness of the velocity of the fluid, usually based on structure functions in the physical space. However, it can also be defined with Fourier high-pass filters. Experimental and numerical simulations suggest that the two approaches do not always give the same results. Our purpose is to compare them from the analytical point of view of functions. We do that by studying generalizations of Riemann's non-differentiable function, yielding computations that are related to some classical problems in Fourier analysis. The conclusion is that the result strongly depends on regularity. To visualize this, we establish an analogy between these generalizations and the influence of viscosity in turbulent flows. This article is motivated by the mathematical works on the multifractal formalism and the discovery of Riemann's non-differentiable function as a trajectory of polygonal vortex filaments.

Interacting helical traveling waves for the Gross–Pitaevskii equation

We consider the three-dimensional Gross–Pitaevskii equation i\partial_t \psi +\Delta \psi+(1-|\psi|^2)\psi=0\quad \text{for $\psi\colon\mathbb{R}\times \mathbb{R}^3 \rightarrow \mathbb{C}$} and construct traveling wave solutions to this equation. These are solutions of the form \psi(t,x)=u(x_1,x_2,x_3-Ct) with a velocity C of order \varepsilon\lvert\log\varepsilon\rvert for a small parameter \varepsilon>0 . We build two different types of solutions. For the first type, the functions u have a zero-set (vortex set) close to a union of n helices for n\geq 2 and near these helices u has degree 1 . For the second type, the functions u have a vortex filament of degree -1 near the vertical axis e_3 and n\geq 4 vortex filaments of degree +1 near helices whose axis is e_3 . In both cases the helices are at a distance of order 1/(\varepsilon\sqrt{\lvert\log\varepsilon\rvert}) from the axis and are solutions to the Klein–Majda–Damodaran system, supposed to describe the evolution of nearly parallel vortex filaments in ideal fluids. Analogous solutions have been constructed recently by the authors for the stationary Gross–Pitaevskii equation, namely the Ginzburg–Landau equation. To prove the existence of these solutions we use the Lyapunov–Schmidt method and a subtle separation between even and odd Fourier modes of the error of a suitable approximation.

Hasimoto Maps for Nonlinear Schrödinger Equations in Minkowski Space

In this paper, we study the vortex filament flow for timelike and spacelike curves in Minkowski 3-space. The vortex filament flow equations of the timelike and the spacelike curves are equivalent to the nonlinear Schrödinger equation and the heat equation, respectively. As a consequentce, we prove that a soliton of the nonlinear Schrödinger equations of the timelike curve gives a solution of a traveling wave on a line at infinity. Also, we study a solution of a traveling wave of the nonlinear Schrödinger equations of the spacelike curve in terms of a new complex frame. Finally, we discuss the method to find the exact shape of the timelike and the spacelike curves from the vortex filament by solving the Frenet vectors of these curves and provide applications to illustrate the method.

A Proper-Orthogonal-Decomposition (POD) Study of the Wake Characteristics behind a Wind Turbine Model

A comprehensive study was performed to analyze turbine wake characteristics by using a Proper-Orthogonal-Decomposition (POD) method to identify the dominant flow features from a comprehensive experimental database. The wake flow characteristics behind a typical three-bladed horizontal-axis wind turbine (HAWT) were measured in a large-scale wind tunnel with a scaled turbine model placed in a typical offshore Atmospheric Boundary Layer (ABL) wind under a neutral stability condition. A high-resolution Particle Image Velocimetry (PIV) system was used to achieve detailed flow field measurements to characterize the turbulent flows and wake vortex structures behind the turbine model. Statistically averaged measurements revealed the presence of the characteristic helical-tip vortex filament along with a unique secondary vortex filament emanating from 60% of the blade span measured from the hub. Both filaments breakup in the near-wake region (~0.6 rotor diameter downstream) to form shear layers, contrary to previous computational and experimental observations in which vortex filaments break up in the far wake. A Proper-Orthogonal-Decomposition (POD) analysis, based on both velocity and vorticity-based formulations, was used to extract the coherent flow structures, predominantly comprised of tip and midspan vortex elements. The reconstructions showed coherence in the flow field prior to the vortex breakup which subsequently degraded in the turbulent shear layer. The accuracy of the POD reconstructions was validated qualitatively by comparing the prediction results between the velocity and vorticity-based formulations as well as the phase-averaged PIV measurement results. This early vortex breakup was attributed to the reduced pitch between consecutive helical turns, the proximity between midspan filaments and blade tips as well as the turbulence intensity of the incoming boundary layer wind.