Vortex Sheet Strength Research Articles

Motion of a current-vortex sheet in the magnetic Kelvin-Helmholtz instability.

In this paper, we consider the Kelvin-Helmholtz instability in the magnetohydrodynamic flow. The motion of the interface is described by a current-vortex sheet. We examine the linear stability of the current-vortex sheet model and determine the growth rate of the interface. The interface is linearly stable for M_{A}<2 where M_{A} represents the Alfvén Mach number. It is found that the interface is linearly unstable in the limit of the critical Alfvén Mach number M_{A}=2, due to resonance of eigenvalues. We perform numerical simulations for the current-vortex sheet for both regimes of M_{A}<2 and M_{A}>2. The numerical results show the stabilizing effects of the magnetic field on the evolution of the current-vortex sheet when the magnetic field is sufficiently large. For the regime M_{A}<2, the sheet oscillates both longitudinally and transversely and the transverse surface wave is pronounced for a large M_{A}. Remarkably, the interface is nonlinearly unstable for M_{A}≈2, for M_{A}<2, which may be due to the propagation of surface waves. For the regime M_{A}>2, the roll-up of the spiral is weakened and the spiral is more pinched and stretched for smaller M_{A}. A comparison of the unstable evolutions of large and small values of M_{A} shows significant differences of the magnetic field and vortex sheet strength.

A volume-of-fluid vortex sheet method for multiphase flows

An Eulerian method to solve the vortex sheet dynamics of an interface captured by a Volume-of-Fluid (VoF) transport scheme is presented. The VoF scalar, vortex sheet strength, and interfacial area are numerically transported using an unsplit geometric scheme and source terms are evaluated with finite differences. Several methods are proposed to generate the self-induced vortex sheet velocities and the methods are assessed based on their computational feasibility and solution verification properties. A grid convergence study is conducted to evaluate the convergence properties of the VoF vortex sheet method and several classical hydrodynamic instabilities are presented to highlight the capabilities of the method in 2D and 3D. Results are compared against prior literature and Direct Numerical Simulations.

Numerical algorithms for water waves with background flow over obstacles and topography

We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

Planar potential flow on Cartesian grids

Potential flow has many applications, including the modelling of unsteady flows in aerodynamics. For these models to work efficiently, it is best to avoid Biot–Savart interactions. This work presents a grid-based treatment of potential flows in two dimensions and its use in a vortex model for simulating unsteady aerodynamic flows. For flows consisting of vortex elements, the treatment follows the vortex-in-cell approach and solves the streamfunction–vorticity Poisson equation on a Cartesian grid after transferring the circulation from the vortices onto the grid. For sources and sinks, an analogous approach can be followed using the scalar potential. The combined velocity field due to vortices, sinks and sources can then be obtained using the Helmholtz decomposition. In this work, we use several key tools that ensure the approach works on arbitrary geometries, with and without sharp edges. First, the immersed boundary projection method is used to account for bodies in the flow and the resulting body-forcing Lagrange multiplier is identified as the bound vortex sheet strength. Second, sharp edges are treated by decomposing the vortex sheet strength into a singular and non-singular part. To enforce the Kutta condition, the non-singular part can then be constrained to remove the singularity introduced by the sharp edge. These constraints and the Poisson equation are formulated as a saddle-point system and solved using the Schur complement method. The lattice Green's function is used to efficiently solve the discrete Poisson equation with unbounded boundary conditions. The method and its accuracy are demonstrated for several problems.

Dissipative Euler Flows for Vortex Sheet Initial Data without Distinguished Sign

AbstractWe construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our initial datum has vorticity concentrated on a simple closed curve in a suitable Hölder space and the vorticity may not have a distinguished sign. Our solutions are obtained by means of convex integration; they are smooth outside a “turbulence” zone which grows linearly in time around the vortex sheet. As a by‐product, this approach shows how the growth of the turbulence zone is controlled by the local energy inequality and measures the maximal initial dissipation rate in terms of the vortex sheet strength. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Boundary layer vortex sheet evolution around an accelerating and rotating cylinder

Abstract

Axisymmetric contour dynamics for buoyant vortex rings

The present work uses a reduced-order model to study the motion of a buoyant vortex ring with non-negligible core size. Buoyancy is considered in both non-Boussinesq and Boussinesq situations using an axisymmetric contour dynamics formulation. The density of the vortex ring differs from that of the ambient fluid, and both densities are constant and conserved. The motion of the ring is calculated by following the boundary of the vortex core, which is also the interface between the two densities. The velocity of the contour comes from a combination of a specific continuous vorticity distribution within its core and a vortex sheet on the core boundary. An evolution equation for the vortex sheet is derived from the Euler equation, which simplifies considerably in the Boussinesq limit. Numerical solutions for the coupled integro-differential equations are obtained. The dynamics of the vortex sheet and the formation of two possible singularities, including singularities in the curvature and the shock-like profile of the vortex sheet strength, are discussed. Three dimensionless groups, the Atwood, Froude and Weber numbers, are introduced to measure the importance of physical effects acting on the motion of a buoyant vortex ring.

Vortex sheet roll-up revisited

The classical problem of roll-up of a two-dimensional free inviscid vortex sheet is reconsidered. The singular governing equation brings with it considerable difficulty in terms of actual calculation of the sheet dynamics. Here, the sheet is discretized into segments that maintain it as a continuous object with curvature. A model for the self-induced velocity of a finite segment is derived based on the physical consideration that the velocity remain bounded. This allows direct integration through the singularity of the Birkhoff–Rott equation. The self-induced velocity of the segments represents the explicit inclusion of stretching of the sheet and thus vorticity transport. The method is applied to two benchmark cases. The first is a finite vortex sheet with an elliptical circulation distribution. It is found that the self-induced velocity is most relevant in regions where the curvature and the sheet strength or its gradient are large. The second is the Kelvin–Helmholtz instability of an infinite vortex sheet. The critical time at which the sheet forms a singularity in curvature is accurately predicted. As observed by others, the vortex sheet strength forms a finite-valued cusp at this time. Here, it is shown that the cusp value rapidly increases after the critical time and is the impetus that initiates the roll-up process.

Generalized Contour Dynamics: A Review

Contour dynamics is a computational technique to solve for the motion of vortices in incompressible inviscid flow. It is a Lagrangian technique in which the motion of contours is followed, and the velocity field moving the contours can be computed as integrals along the contours. Its best-known examples are in two dimensions, for which the vorticity between contours is taken to be constant and the vortices are vortex patches, and in axisymmetric flow for which the vorticity varies linearly with distance from the axis of symmetry. This review discusses generalizations that incorporate additional physics, in particular, buoyancy effects and magnetic fields, that take specific forms inside the vortices and preserve the contour dynamics structure. The extra physics can lead to time-dependent vortex sheets on the boundaries, whose evolution must be computed as part of the problem. The non-Boussinesq case, in which density differences can be important, leads to a coupled system for the evolution of both mean interfacial velocity and vortex sheet strength. Helical geometry is also discussed, in which two quantities are materially conserved and whose evolution governs the flow.

Vortex Sheet Strength in the Sears, Küssner, Theodorsen, and Wagner Aerodynamics Problems

The seminal aerodynamics literature provides analytic predictions of the loads due to sinusoidal gusts (Sears and von Kármán), sharp-edged transverse gusts (Küssner), sinusoidal motions (Theodorsen), and step-change airfoil motions (Wagner). Although these workers determined the overall loads (circulation, lift, and moment), they did not explicitly derive the vortex sheet strength (which is used to compute these loads). Simply put, vortex sheet strength is the velocity difference above/below the airfoil, so it is related to the pressure distribution and therefore the loads. Knowledge of the vortex sheet strength is important because, to extend this seminal theory to include nonlinearities such as thickness–load coupling, viscous–load coupling, or boundary-layer separation, one needs to inform these calculations with the vortex sheet strength. The main contribution of this paper is a method to theoretically predict the vortex sheet strength in the seminal unsteady aerodynamics problems of Sears, Küssner, Theodorsen, and Wagner. These theoretical calculations are enabled by developing a numerical method for calculating the required Fourier coefficients. In addition, a unified presentation of linear unsteady aerodynamics theory is contributed, and examples are provided to illustrate the vortex sheet strength in each of the four seminal problems.

Nonlinear Dynamics of Non-uniform Current-Vortex Sheets in Magnetohydrodynamic Flows

A theoretical model is proposed to describe fully nonlinear dynamics of interfaces in two-dimensional MHD flows based on an idea of non-uniform current-vortex sheet. Application of vortex sheet model to MHD flows has a crucial difficulty because of non-conservative nature of magnetic tension. However, it is shown that when a magnetic field is initially parallel to an interface, the concept of vortex sheet can be extended to MHD flows (current-vortex sheet). Two-dimensional MHD flows are then described only by a one-dimensional Lagrange parameter on the sheet. It is also shown that bulk magnetic field and velocity can be calculated from their values on the sheet. The model is tested by MHD Richtmyer–Meshkov instability with sinusoidal vortex sheet strength. Two-dimensional ideal MHD simulations show that the nonlinear dynamics of a shocked interface with density stratification agrees fairly well with that for its corresponding potential flow. Numerical solutions of the model reproduce properly the results of the ideal MHD simulations, such as the roll-up of spike, exponential growth of magnetic field, and its saturation and oscillation. Nonlinear evolution of the interface is found to be determined by the Alfven and Atwood numbers. Some of their dependence on the sheet dynamics and magnetic field amplification are discussed. It is shown by the model that the magnetic field amplification occurs locally associated with the nonlinear dynamics of the current-vortex sheet. We expect that our model can be applicable to a wide variety of MHD shear flows.

Airfoil in a high amplitude oscillating stream

A combined theoretical and experimental investigation was carried out with the objective of evaluating theoretical predictions relating to a two-dimensional airfoil subjected to high amplitude harmonic oscillation of the free stream at constant angle of attack. Current theoretical approaches were reviewed and extended for the purposes of quantifying the bound, unsteady vortex sheet strength along the airfoil chord. This resulted in a closed form solution that is valid for arbitrary reduced frequencies and amplitudes. In the experiments, the bound, unsteady vortex strength of a symmetric 18 % thick airfoil at low angles of attack was measured in a dedicated unsteady wind tunnel at maximum reduced frequencies of 0.1 and at velocity oscillations less than or equal to 50 %. With the boundary layer tripped near the leading edge and mid-chord, the phase and amplitude variations of the lift coefficient corresponded reasonably well with the theory. Near the maximum lift coefficient overshoot, the data exhibited an additional high-frequency oscillation. Comparisons of the measured and predicted vortex sheet indicated the existence of a recirculation bubble upstream of the trailing edge which sheds into the wake and modifies the Kutta condition. Without boundary layer tripping, a mid-chord bubble is present that strengthens during flow deceleration and its shedding produces a dramatically different effect. Instead of a lift coefficient overshoot, as per the theory, the data exhibit a significant undershoot. This undershoot is also accompanied by high-frequency oscillations that are characterized by the bubble shedding. In summary, the location of bubble and its subsequent shedding play decisive roles in the resulting temporal aerodynamic loads.

Some aspects relating to the accuracy of high-order panel methods

In this study we have examined two numerical algorithms, based upon the high-order panels approach, to verify and demonstrate that a spurious solution is a direct result of a low-order scheme accuracy that violates the conservation of circulation. The first algorithm is based upon the parabolic Lagrange interpolation and the second one is based on the same parabolic Lagrange interpolation for the vortex sheet strength where the geometry of the body/foil is evaluated using a periodic cubic spline, mainly because the denominator in the integrand of the Cauchy algorithm is the main source of the numerical error in the scheme. Good agreement has been found between the computational and analytical results and a spurious free solution has been obtained for the high-order schemes, except for a spurious-like solution in the case of an under resolved problem.

Dependence of Time-periodic Vortex Sheets with Surface Tension on Mean Vortex Sheet Strength

In recent work, the authors have computed time-periodic solutions of the vortex sheet with surface tension in the spatially periodic setting. In these prior results, the mean vortex sheet strength was taken to be zero, so that there is no net shear across the fluid interface. In the current work, we explore the effect of allowing such a net shear, finding time-periodic vortex sheets that overturn without rolling up. We also find that resonances between Fourier modes lead to disconnections in the bifurcation curves that describe a two-parameter family of time-periodic solutions.

Traveling waves from the arclength parameterization: Vortex sheets with surface tension

We study traveling waves for the vortex sheet with surface tension. We use the angle-arclength description of the interface rather than Cartesian coordinates, and we utilize an arclength parameterization as well. In this setting, we make a new formulation of the traveling wave ansatz. For this problem, it should be possible for traveling waves to overturn, and notably, our formulation does allow for waves with multi-valued height. We prove that there exist traveling vortex sheets with surface tension bifurcating from equilibrium. We compute these waves by means of a quasi-Newton iteration in Fourier space; we find continua of traveling waves bifurcating from equilibrium and extending to include overturning waves, for a variety of values of the mean vortex sheet strength.

Singularity formation and nonlinear evolution of a viscous vortex sheet model

We study Dhanak's model [J. Fluid Mech. 269, 265 (1994)]10.1017/S0022112094001552 of a viscous vortex sheet in the sharp limit, to investigate singularity formations and present nonlinear evolutions of the sheets. The finite-time singularity does not disappear by giving viscosity to the vortex sheet, but is delayed. The singularity in the sharp viscous vortex sheet is found to be different from that of the inviscid sheet in several features. A discontinuity in the curvature is formed in the viscous sheet, similarly as the inviscid sheet, but a cusp in the vortex sheet strength is less sharpened by viscosity. Exponential decay of the Fourier amplitudes is lost by the formation of the singularity, and the amplitudes of high wavenumbers exhibit an algebraic decay, while in the inviscid vortex sheet, the algebraic decay of the Fourier amplitudes is valid from fairly small wavenumbers. The algebraic decay rate of the viscous vortex sheet is approximately −2.5, independent of viscosity, which is the same rate as the asymptotic analysis of the inviscid sheet. Results for evolutions of the regularized vortex sheets show that the roll-up is weakened by viscosity, and the regularization parameter has more significant effects on the fine-structure of the core than does viscosity.

The attraction between a flexible filament and a point vortex

AbstractWe determine the inviscid dynamics of a point vortex in the vicinity of a flexible filament. For a wide range of filament bending rigidities, the filament is attracted to the point vortex, which generally moves tangentially to it. We find evidence that the point vortex collides with the filament at a critical time, with the separation distance tending to zero like a square root of temporal distance from the critical time. Concurrent with the collision, we find divergences of pressure loading on the filament, filament vortex sheet strength, filament curvature and velocity. We derive the corresponding power laws using the governing equations.

Baroclinic vortex sheet production by shocks and expansion waves

Vortex sheet production by shocks and expansion waves refracting at a density discontinuity was examined and compared using an analytical solution and numerical simulations. The analytical solution showed that with a small exception, vortex sheet strength is generally stronger in fast/slow shock refractions. In contrast, expansion waves generated a stronger vortex sheet in slow/fast refractions. This difference results in larger vorticity deposited by shocks in fast/slow refractions and by expansion waves in slow/fast refractions. Shock refractions become irregular and the analytical solution fails when either incident, transmitted or reflected shock, exceeded the angle limit for an attached shock. To investigate vortex sheet production outside the range of analytical solutions and to verify the applicability of the planar-interface analytical solution to a curved interface, shock refraction through a sinusoidal interface was numerically simulated in the shock frame of reference. It is found that variation in the local incidence angle along the curved interface creates pressure waves that affect the level of deposited vorticity. This contributes to the difference between predictions from local analysis and numerical computation. Furthermore, an interesting behavior of the shock and expansion wave-deposited vorticity in supersonic ramp flow was discovered. When the high- and low-density streams were swapped, while keeping the incident flow Mach numbers constant, a vortex sheet of equal magnitude but of opposite sign was generated.

Coriolis effects on fingering patterns under rotation

The development of immiscible viscous fingering patterns in a rotating Hele-Shaw cell is investigated. We focus on understanding how the time evolution and the resulting morphologies are affected by the action of the Coriolis force. The problem is approached analytically and numerically by employing a vortex sheet formalism. The vortex sheet strength and a linear dispersion relation are derived analytically, revealing that the most relevant Coriolis force contribution comes from the normal component of the averaged interfacial velocity. It is shown that this normal velocity, uniquely due to the presence of the Coriolis force, is responsible for the complex-valued nature of the linear dispersion relation making the linear phases vary with time. Fully nonlinear stages are studied through intensive numerical simulations. A suggestive interplay between inertial and viscous effects is found, which modifies the dynamics, leading to different pattern-forming structures. The inertial Coriolis contribution plays a characteristic role: it generates a phase drift by deviating the fingers in the sense opposite to the actual rotation of the cell. However, the direction and intensity of finger bending is predominantly determined by viscous effects, being sensitive to changes in the magnitude and sign of the viscosity contrast. The finger competition behavior at advanced time stages is also discussed.

Impact of a vortex ring on a density interface using a regularized inviscid vortex sheet method

A new, fully three-dimensional, vortex-in-cell method designed to follow the unsteady motion of inviscid vortex sheets with or without small (Boussinesq) density discontinuities is presented. As is common in front-tracking methods, the vortex sheet is described by a moving, unstructured mesh consisting of points connected by triangular elements. Each element carries scalar-valued circulations on its three edges, which can be used to represent any tangent vector value and in the present method represent the element’s vorticity. As the interface deforms, nodes and elements are added and removed to maintain the resolution of the sheet and of the vortex sheet strength. The discretization and remeshing methods allow automatic, near-perfect conservation of circulation despite repeated stretching and folding of the interface. Results are compared with previous experiments and simulations. Similarities are observed between the present simulations and experiments of a vortex ring impacting a wall.