Vortex Sheet Research Articles

Spiral wind-up of vortex sheets

A model of the spiral wind-up of a vortex sheet by the flow induced by a perpendicular vortex is developed, first in the Euler limit of zero viscosity, then taking account of viscous diffusion. The wind-up of a vortex sheet by a propagating vortex pair is also considered; in this case, the vorticity grows exponentially near the two saddle points of the flow in the moving frame of the vortex pair. Estimates are obtained for the maximum vorticity ω m and maximum enstrophy Ω m attained through this process in terms of the wind-up Reynolds number Re and the initial dimensionless sheet thickness δ 0 . This paper is dedicated to Andrew Soward, whose sustained research in Dynamo Theory and Fluid Mechanics has been brilliantly inspiring for more than 50 years. Happy 80th, Andrew!

Caudal peduncle-inspired two-degree-of-freedom elastic coupling fin propulsion method

Marine animals orchestrate the swimming process through the coordinated interplay of body musculature, the caudal peduncle, and the caudal fin. However, understanding the coordinated action of these components to achieve high propulsive performance remains a significant challenge. The study proposes a self-propulsive physical model with two-degree-of-freedom (DoF) elastic coupling inspired by the caudal peduncle, where the caudal peduncle exhibits spring-like behaviors influencing the tail's motion along heave/pitch directions. The complex nonlinear fluid–structure interaction issues are addressed via the nonlinear vortex sheet method. The study primarily compares the propulsive performance of the two-DoF elastic coupling caudal fin model with the pitch caudal fin model. Numerical results show that the peak efficiency of the proposed model is nearly eight times that of the pitch caudal fin model. Additionally, the study reveals that the high-propulsive mechanism lies in generating the figure of a butterfly phase diagram for the hydrodynamic forces and exploiting vortices to decrease energy consumption. These findings offer novel perspectives for the future design of high-efficiency underwater robots.

A numerical study of vortex nucleation in 2D rotating Bose–Einstein condensates

This article implements a numerical method for the minimization under constraints of a discrete energy modelling multicomponents rotating Bose–Einstein condensates in the regime of strong confinement and with rotation. Moreover, this method allows to consider both segregation and coexistence regimes between the components. The method includes a discretization of a continuous energy in space dimension 2 and a gradient algorithm with adaptive time step and projection for the minimization. It is well known that, depending on the regime, the minimizers may display different structures, sometimes with vorticity (from singly quantized vortices, to vortex sheets and giant holes). The goal of this paper is to study numerically the structures of the minimizers. In order to do so, we introduce a numerical algorithm for the computation of the indices of the vortices, as well as an algorithm for the computation of the indices of vortex sheets. Several computations are carried out, to illustrate the efficiency of the method, to cover different physical cases, to validate recent theoretical results as well as to support conjectures. Moreover, we compare this method with an alternative method from the literature.

A Josephson–Anderson relation for drag in classical channel flows with streamwise periodicity: Effects of wall roughness

The detailed Josephson–Anderson relation equates the instantaneous work by pressure drop over any streamwise segment of a general channel and the wall-normal flux of spanwise vorticity spatially integrated over that section. This relation was first derived by Huggins for quantum superfluids, but it holds also for internal flows of classical fluids and for external flows around solid bodies, corresponding there to relations of Burgers, Lighthill, Kambe, Howe, and others. All of these prior results employ a background potential Euler flow with the same inflow/outflow as the physical flow, just as in Kelvin's minimum energy theorem, so that the reference potential incorporates information about flow geometry. We here generalize the detailed Josephson–Anderson relation to streamwise periodic channels appropriate for numerical simulation of classical fluid turbulence. We show that the original Neumann b.c. used by Huggins for the background potential creates an unphysical vortex sheet in a periodic channel, so that we substitute instead Dirichlet b.c. We show that the minimum energy theorem still holds and our new Josephson–Anderson relation again equates work by pressure drop instantaneously to integrated flux of spanwise vorticity. The result holds for both Newtonian and non-Newtonian fluids and for general curvilinear walls. We illustrate our new formula with numerical results in a periodic channel flow with a single smooth bump, which reveals how vortex separation from the roughness element creates drag at each time instant. Drag and dissipation are thus related to vorticity structure and dynamics locally in space and time, with important applications to drag-reduction and to explanation of anomalous dissipation at high Reynolds numbers.

Center vortex geometry at finite temperature

The geometry of center vortices is studied in SU(3) gauge theory at finite temperature to capture the key structural changes that occur through the deconfinement phase transition. Visualizations of the vortex structure in temporal and spatial slices of the lattice reveal a preference for the vortex sheet to align with the temporal dimension above the critical temperature. This is quantified through a correlation measure. A collection of vortex statistics, including vortex and branching point densities, and vortex path lengths between branching points, are analyzed to highlight internal shifts in vortex behavior arising from the loss of confinement. We find the zero-temperature inclination of branching points to cluster at short distances vanishes at high temperatures, embodying a rearrangement of branching points within the vortex structure. These findings establish the many aspects of center vortex geometry that characterize the phase transition in pure gauge theory. Published by the American Physical Society 2024

Comparison of Libration- and Precession-Driven Flows: From Linear Responses to Broadband Dynamics

Libration and precession are different body forces that are ubiquitous in many rapidly rotating systems, particularly in geophysical and astrophysical flows. Libration is a modulation of the background rotation magnitude, whereas precession is a modulation of the background rotation direction. Assessing the consequences of these body forces in large-scale flows is challenging. The Ekman number, the ratio of the rotation time scale to the viscous time scale quantifying the rotation speed, is extremely small, leading to extremely thin and intense shear layers in the flows even when the amplitudes of the body forces are very small. We consider the consequences of libration and precession numerically in a geometrically simple container, a cube, which lends itself to very efficient, accurate, and robust numerical treatment, with the axis of rotation passing through opposite vertices, so that all walls of the cube are at oblique angles to the rotation axis. This results in the geometric focusing of inertial wavebeams reflecting off the walls, whereby the energy density of the wavebeams increases along with the magnitude of their wavevector. The nature of this focusing depends on the forcing frequency but not on the body force. In the inviscid setting, wavebeams form infinitesimally thin vortex sheets, and their energy density becomes unbounded upon focusing. We present linear inviscid ray tracing to set the scene for the focusing of wavebeams and then consider viscous problems at an Ekman number that is typical of current state-of-the-art laboratory experiments. We begin by considering the linear responses, which are comprised of focusing viscous shear layers, of which their details are mostly captured via ray tracing, and particular solutions accounting for the body forces. These have complicated spatio-temporal structures, which differ for libration and precession. Increasing the forcing amplitude from zero introduces nonlinear interactions, enhances the focusing effects via vortex tilting and stretching when the shear layers reflect at the walls, and also introduces temporal superharmonics and a mean flow. When the magnitude of the mean flow is within a few percent of the magnitude of the instantaneous flow, instabilities breaking the spatio-temporal symmetries set in. These are localized in the oscillatory boundary layers where the reflections are concentrated and introduce broadband dynamics in the boundary layers, with additional inertial wavebeams emitted into the interior. The details again depend on the specifics of the body forces.

A high-resolution numerical investigation of unsteady wake vortices for coaxial rotors in hover

High-resolution numerical simulations for wake vortical flows have long been a challenge in rotor aerodynamics. A novel spectrum-optimized sixth-order Weighted Essentially Non-Oscillatory (WENO) scheme is proposed to discretize inviscid fluxes on moving overset grids, and the Improved Delayed Detached Eddy Simulation (IDDES) is employed to resolve turbulent vortices. The integration of these methods facilitates a comprehensive numerical investigation into the unsteady vortical flows over coaxial rotors in hover. The results highlight the substantial improvement in numerical resolution, in terms of both spatial structure and temporal evolution of unsteady multiscale wake vortices. Coaxial rotors in hover manifest three primary scales of wake vortex structures: (A) the helical evolution of primary blade tip vortices and the periodic occurrence of strong Blade-Vortex Interactions (BVI); (B) the continuous shedding of small-scale horseshoe-shaped vortices from the trailing edges of rotor blades, forming the vortex sheets; (C) the emergence of small-scale secondary vortex braids induced by interactions between rotor tip vortices and the vortex sheets. These vortex structures and their interactions cause high-frequency oscillations in rotor disk loads and induce unsteady perturbations in the local flow field. Interactions among these primary vortices, coupled with the generation of secondary vortices, result in the dissipation, distortion, and breakup of the rotor tip vortices, ultimately forming a vortex soup. Notably, a substantial quantity of seemingly weak small-scale secondary vortex braids significantly contribute to energy dissipation during the evolution of wake vortices for coaxial rotors in hover.

Investigation into the Vortex Evolution characteristics of the Francis Turbine During the Runaway Process

Abstract The runaway process is a dynamic transition process that gradually deviates from the optimal operating condition. The flow passage components dissipate all of the energy corresponding to the hydraulic turbine’s head, causing the internal flow of the hydraulic turbine to gradually deteriorate and produce a large-scale vortex structure during the runaway process. A model Francis turbine is used as the research object in this paper to understand the evolution characteristics of an unsteady vortex during a runaway. The runaway process is investigated using high-precision simulation technology of gas-liquid two-phase flow, and the runaway speed is obtained, consistent with the experimental results. The results show that the cavitation volume increases as the rotating speed increases. The large incidence angle between the inlet flow and the blade causes large-scale flow separation in the runner, which leads to forming of a sheet vortex band near the hub of the runner hub and a columnar vortex in the blade passage of the runner blade suction surface. As the runaway process continues, the flow separation in the runner intensifies and the induced vortex size increases, obstructing the runner’s passage. In addition, pressure signal analysis reveals that the pressure fluctuation caused by the sheet vortex band is low-frequency. On the other hand, the pressure fluctuation caused by the columnar vortex is frequent. These two types of vortex structures with high-energy pressure fields degrade the flow in the runner and reduce the hydraulic performance of the turbine.

On a stabilization of the Ingard-Myers impedance boundary condition and its time domain implementation

It has been well-known that the Ingard-Myers impedance condition, while simple to apply, is subject to the hydrodynamic Kelvin-Helmholtz-type instability due to its use of a vortex sheet in modeling the flow at the liner boundary. Recently, in the development of a time domain boundary element method for acoustic scattering by treated surfaces, it was found that by neglecting a certain second-order spatial derivative term in the Ingard-Myers formulation, the hydrodynamic instability can be avoided. The present paper aims to provide further analysis of this modified condition, hereby referred to as the Truncated Ingard-Myers Impedance Boundary Condition (TIMIBC). It will be shown, based on the dispersion relations of linear waves, that the instability intrinsic to the Ingard-Myers condition is eliminated in the proposed new formulation. Quantitative assessments on the accuracy of the TIMIBC for scattering of acoustic waves by lined surfaces are carried out, and its effectiveness is demonstrated by a numerical example. It is found that the TIMIBC provides a good approximation to the original Ingard-Myers condition for flows of low to mid subsonic Mach numbers. As such, the proposed TIMIBC can offer a practical solution for overcoming the intrinsic instability associated with the Ingard-Myers condition. Moreover, time domain implementation of the TIMIBC is also discussed and illustrated with a numerical example using a finite difference scheme. In particular, a minimization procedure for finding the poles and coefficients of a broadband multipole expansion for the impedance function is formulated by which, unlike the commonly used vector-fitting method, passivity of the model is ensured.

The time evolution of the mother body of a planar uniform vortex moving in an inviscid fluid

The time evolution of the mother body of a planar, uniform vortex that moves in an incompressible, inviscid fluid is investigated. The vortex is isolated, so that its motion is just due to self-induced velocities. Its mother body is defined as the part internal to the vortex of the singular set of the Schwarz function of its boundary. In the present analysis, it is an arc of curve (branch cut), starting and ending in the two internal branch points of this function, across any point of which the Schwarz function experiences a finite jump. By looking at the mother body from the outside of the vortex, it behaves as a vortex sheet having density of circulation given by the jump of the Schwarz function. Its name (mother body) is taken from Geophysics, and it is here used due to its property of generating, outside the vortex and on its boundary, the same velocity as the vortex itself. The shape of the branch cut and the jump of the Schwarz function across any point of it change in time, by following the motion of the vortex boundary. As it happens for a physical vortex sheet, the mother body is not a material line, so that it does not move according to the velocities induced by the vortex. In the present paper, the cut shape, the above jumps, as well as the cut velocities are deduced from the time evolution equation of the Schwarz function. Numerical experiments, carried out by building the branch cut and calculating the limit values of the Schwarz function on its sides during the vortex motion, confirm the analytical calculations. Some global quantities (circulation, first and second order moments) are here rewritten as integrals on the cut, and their conservation during the vortex motion is analytically and numerically verified. Indeed, the numerical simulations show that they behave in the same way as their classical contour dynamics forms, written in terms of integrals on the vortex boundary. This proves that the shape of the cut, as well as the limit values of the Schwarz function on its sides, are correctly calculated during the motion.

Combined Theodorsen and Sears theory: experimental validation and modification

The response of airfoils to unsteady disturbances is a classic problem in the aerodynamics field. Many theoretical models have been proposed in the past to predict the unsteady aerodynamic forces of airfoils. However, these theories focused on individual airfoil motions or incoming flow disturbances, while the theoretical models for multiple disturbances still need to be developed. In this study, a theoretical model to predict the aerodynamic force of an oscillating airfoil encountering vertical gust is derived from a linear combination of Theodorsen's and Sears’ theories. Experimental investigations involving a two-dimensional pitching airfoil encountering a sinusoidal vertical gust are carried out to examine the proposed theory. It is found that the theory effectively captures the trends in the unsteady lift of airfoils subjected to dual disturbances. However, it tends to overestimate the lift amplitude. Notably, when a quasi-steady correction is applied to the theory, the prediction accuracy is greatly improved. The theory correction agrees well with experiment at small pitching frequencies, while deviations exist at higher pitching frequencies. The temporal evolution of the flow velocity reveals that the velocity disturbance induced by the coupled disturbance around the airfoil conforms to the linear superposition of the velocities induced by each individual disturbance, consistent with the prediction of the vortex sheet model. As the pitching frequency increases, significant nonlinear effects appear near the trailing edge of the airfoil, which may be one key factor for the disparities between the theoretical predictions and the experimental lift at higher pitching frequencies.

Extended Dynamic Mode Decomposition with Invertible Dictionary Learning

The Koopman operator has received attention for providing a potentially global linearization representation of the nonlinear dynamical system. To estimate or control the original system, the invertibility problem is introduced into the data-driven modeling, i.e., the observables are required to be reconstructed the original system’s states. Existing methods cannot solve this problem perfectly. Only linear or nonlinear but lossy reconstruction can be achieved. This paper proposed a novel data-driven modeling approach, denoted as the Extended Dynamic Mode Decomposition with Invertible Dictionary Learning (EDMD-IDL) to address this issue, which can be interpreted as a further extension of the classical Extended Dynamic Mode Decomposition (EDMD). The Invertible Neural Network (INN) is introduced in the proposed method, where its inverse process provides the explicit inverse on the dictionary functions, thus allowing the nonlinear and lossless reconstruction. An iterative algorithm is designed to solve the extended optimization problem defined by the Koopman operator and INN by combining the optimization algorithm based on the gradient descent and the classical EDMD method, making the method successfully obtain the finite-dimensional approximation of the Koopman operator. The method is tested on various canonical nonlinear dynamical systems and is shown that the predictions obtained in a linear fashion and the ground truth match well over the long-term, where only the initial status is provided. Comparison experiments highlight the superiority of the proposed method over the other EDMD-based methods. Notably, a typical example in fluid dynamics, cylinder wake, illustrates the potential of the method to be further extended to the high-dimensional system with tens of thousands of states. By combining the Proper Orthogonal Decomposition technique, nontrivial Kármán vortex sheet phenomenon is perfectly reconstructed. Our proposed method provides a new paradigm for solving the finite-dimensional approximation of the Koopman operator and applying it to data-driven modeling.

Vorticity dynamics at partial-slip boundaries

In this paper we discuss the dynamics of vorticity at partial-slip boundaries. We consider the total vector circulation, which includes both the total vorticity of the fluid and the slip velocity at the boundary (the interface vortex sheet). The generation of vector circulation is an inviscid process, which does not depend on either viscosity or the slip length at the boundary. Vector circulation is generated by the inviscid relative acceleration between the fluid and the solid, due to either tangential pressure gradients or tangential acceleration of the partial-slip wall. While the slip length does not affect the creation of vector circulation, it governs how vector circulation is distributed between the total vorticity of the fluid and the interface vortex sheet. Specifically, the partial-slip boundary condition prescribes the ratio between boundary vorticity and the strength of the interface vortex sheet, and the viscous boundary flux transfers vector circulation between the interface vortex sheet and the fluid interior to maintain this condition. The interaction between a vortex ring and a partial-slip wall is examined to highlight various aspects of this formulation. For the head-on collision, the quantity of vector circulation diffused into the fluid as secondary vorticity increases as the slip length is decreased, resulting in a stronger secondary vortex and increased rebound of the vortex ring. For the oblique interaction, the extent to which the vortex ring connects to the boundary increases as the slip length is increased.

Corrigendum to “Non-linear singularity formation for circular vortex sheets”

This paper serves as a short corrigendum, describing an error in the derivation of the linearized equation. The main theorems remain correct, but they are based upon an incorrect linearization of the Birkhoff-Rott equation and hence are not directly applicable to the original physical problem.

Numerical investigation on vortex dynamics characteristics of a large vertical centrifugal pump operating in the hump region under cavitation conditions

The large vertical centrifugal pump (LVCP) is widely used in major hydraulic engineering such as cross-region water transfer. This paper focuses on the hydraulic stability of LVCP under extremely operating conditions. The transient numerical simulations of cavitation in LVCP in the hump region are conducted based on delayed detached eddy simulation coupled with a homogeneous cavitation model. The main research is the effect of cavitation on the flow pattern and the vortex dynamics of the LVCP under stall conditions. The results of the numerical investigations show that the variation of the pump performance parameters is consistent with the experimental data under cavitation conditions. Under deep stall conditions, the H drop due to cavitation in the pump can happen earlier. Under the critical cavitation of the stall condition, backflow vortex cavitation, sheet cavitation and separation vortex cavitation can be observed at the impeller inlet. The vortex structures generated at the cavitation tail can result in worse flow patterns in the impeller. The vortex dynamics analysis shows that severe cavitation in the impeller deteriorates the inlet conditions of the vaned diffuser and induces sheet cavitation at the LE of vanes. The cavitations in the vaned diffuser accelerate the production and development of stall vortices and result in increased flow instability. The research in this paper can be helpful to improve the operation stability of LVCP.

Vortex sheet solutions for the Ginzburg–Landau system in cylinders: symmetry and global minimality

Vortex sheet solutions for the Ginzburg–Landau system in cylinders: symmetry and global minimality

Coadjoint orbits of vortex sheets in ideal fluids

We describe coadjoint orbits associated to the motion of codimension one singular vorticities in ideal fluids, e.g. vortex sheets in 3D. We show that these coadjoint orbits can be identified with a certain class of decorated nonlinear Grassmannians, that consist of codimension one submanifolds belonging to an isodrast (i.e., enclosing a given volume, provided their homology class vanishes), endowed with a closed 1-form of a given type. Such isodrasts are defined via an integrable distribution associated to a manifold endowed with a volume form, similar to the isodrastic distribution [35] in symplectic setting. With the choice of a Riemannian metric, the expression of the orbit symplectic form is shown to take a particularly simple expression reminiscent of Darboux coordinates. In general we get coadjoint orbits of the universal central extension of the group of exact volume preserving diffeomorphisms and we give a necessary and sufficient condition under which the central extension is not needed. We also focus on the case in which we get coadjoint orbits of the group of volume preserving diffeomorphisms, which is relevant for the ideal fluid, and discuss how our results extend the class of coadjoint orbits of exact vortex sheets considered in [25].

Dissipation in Onsager’s Critical Classes and Energy Conservation in BV∩L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$BV\\cap L^\\infty $$\\end{document} with and Without Boundary

This paper is concerned with the incompressible Euler equations. In Onsager’s critical classes we provide explicit formulas for the Duchon–Robert measure in terms of the regularization kernel and a family of vector-valued measures {μz}z⊂Mx,t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\\mu _z\\}_z\\subset {\\mathcal {M}}_{x,t}$$\\end{document}, having some Hölder regularity with respect to the direction z∈B1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z\\in B_1$$\\end{document}. Then, we prove energy conservation for Lx,t∞∩Lt1BVx\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty _{x,t}\\cap L^1_t BV_x$$\\end{document} solutions, in both the absence or presence of a physical boundary. This result generalises the previously known case of Vortex Sheets, showing that energy conservation follows from the structure of L∞∩BV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty \\cap BV$$\\end{document} incompressible vector fields rather than the flow having “organized singularities”. The interior energy conservation features the use of Ambrosio’s anisotropic optimization of the convolution kernel and it differs from the usual energy conservation arguments by heavily relying on the incompressibility of the vector field. This is the first energy conservation proof, for a given class of solutions, which fails to simultaneously apply to both compressible and incompressible models, coherently with compressible shocks having non-trivial entropy production. To run the boundary analysis we introduce a notion of “normal Lebesgue trace” for general vector fields, very reminiscent of the one for BV functions. We show that having such a null normal trace is basically equivalent to have vanishing boundary energy flux. This goes beyond the previous approaches, laying down a setup which applies to every Lipschitz bounded domain. Allowing any Lipschitz boundary introduces several technicalities to the proof, with a quite geometrical/measure-theoretical flavour.

Electromagnetic wave-fronts in anisotropic media as discontinuity surfaces. A new mechanical analogy: vortex sheets

Electromagnetic wave-fronts in anisotropic media as discontinuity surfaces. A new mechanical analogy: vortex sheets

A lightweight vortex model for unsteady motion of airfoils

A low-order vortex model has been developed for analysing the unsteady aerodynamics of airfoils. The model employs an infinitely thin vortex sheet in place of the attached boundary layer and a sheet of point vortices for the shed shear layer. The strength and direction of the vortex sheet shed at the airfoil trailing edge are determined by an unsteady Kutta condition. The roll-up of the ambient shear layer is represented by a unique point vortex, which is consistently fed circulation by the last point vortex of the free vortex sheet. The model's dimensionality is reduced by using three tuning parameters to balance representational accuracy and computational efficiency. The performance of the model is evaluated through experiments involving impulsively started and heaving and pitching airfoils. The model accurately captures the dynamics of the development and evolution of the shed vortical structure while requiring minimal computational resources. The validity of the model is confirmed through comparison with experimental force measurements and a baseline unsteady panel method that does not transfer circulation in the free vortex sheet.