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An inviscid vortex shedding model is numerically extended to simulate falling flat plates. The body and vortices separated from the edge of the body are described by vortex sheets. The vortex shedding model has computational limitations when the angle of incidence is small and the free vortex sheet approaches the body closely. These problems are overcome by using numerical procedures such as a method for a near-singular integral and the suppression of vortex shedding at the plate edge. The model is applied to a falling plate of flow regimes of various Froude numbers. For Fr = 0.5 , the plate develops large-scale side-to-side oscillations. In the case of Fr = 1 , the plate motion is a combination of side-to-side oscillations and tumbling and is identified as a chaotic type. For Fr = 1.5 , the plate develops to autorotating motion. Comparisons with previous experimental results show good agreement for the falling pattern. The dependence of change in the vortex structure on the Froude number and its relation with the plate motion is also examined. [ABSTRACT FROM AUTHOR]

In this paper, we study Majda's example for stability of mixed problems introduced in [A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol. 53 (Springer Verlag, 1984)]. After some transformation we analyze the stability of the problem by computing the roots of the Kreiss–Lopatinskiĭ determinant and recover the different cases as in Majda [Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol. 53 (Springer Verlag, 1984)]. Then, we focus on the weakly stable case and prove the a priori estimate of the solution. The proof follows by adapting the approach of Coulombel and Secchi [The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J.53 (2004) 941–1012] for the linear stability of 2D compressible vortex sheets with the simplification introduced by Chen et al. [Linear stability of compressible vortex sheets in two-dimensional elastodynamics, Adv. Math.311 (2017) 18–60]. Compared to [J.-F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J.53 (2004) 941–1012], we improve the a priori energy estimate in that the solution is estimated in suitable weighted Sobolev spaces, anisotropic in the frequency space, whose definition reflects the properties of the associated Lopatinskiĭ determinant. Moreover, we show that this a priori energy estimate is optimal. This very simple example appears useful for the comprehension of the method of proof of the a priori estimate, beyond the technicalities of [R. M. Chen, J. Hu and D. Wang, Linear stability of compressible vortex sheets in two-dimensional elastodynamics, Adv. Math.311 (2017) 18–60; J.-F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J.53 (2004) 941–1012]. [ABSTRACT FROM AUTHOR]

The subject of this work is the development of compact analogs of vortex flows models, which are used in the modeling of vortex structures observed during the flight of an aircraft and the motion of a body in a fluid. In particular, two significant misunderstandings prevailing in this area of science are highlighted. The first misunderstanding is that the stationary motion of fluid parcels in a circle is treated as an inviscid vortex. Therefore, any vortex flow model that does not explicitly contain viscosity is considered to describe inviscid vortex motion. It has been proven that this is not so: the stationary viscous motion of fluid parcels in circular orbits corresponds to the self-balance of one force - the force of viscosity. This conclusion, in an explicit form, was made for the first time. This is very important because it changes our ideas about force balance, where two or more forces of different natures must necessarily be present. Overcoming this misunderstanding opens the way for creating compact analogs of existing models of vortex motions. Along the way, one more - the second general misunderstanding in the field of vortex dynamics was eliminated. Wherever we read it, we can see that the compactness of the vortex flow is associated with the compactness of the vorticity field. This is facilitated by the fact that the equations for vorticity and not for velocity are considered. As a result, except for one or two models of vortices, which correspond to the rotation of the entire space, up to infinity, this violates the fundamental law of physics - the law of conservation and transformation of energy. It is about the fact that, as a second misunderstanding, an error is assumed when calculating the kinetic energy of the vortex flows: the Jacobian in cylindrical (polar) coordinates is not considered. As a result, all the mentioned models of vortex flows, which correspond to the hyperbolic law as their asymptotics in the periphery, have infinite kinetic energy. Certainly, this does not correspond to the formation and evolution of compact vortex structures. Therefore, in this work, based on overcoming the aforementioned misunderstandings, many previously obtained models of compact vortex flows, as well as those obtained for the first time, are presented. In particular, this applies to the turbulent vortex flow during the formation of a vortex sheet, which is a compact analog of the Burgers-Rott vortex - both the classical one corresponding to laminar motion and the one consisting of a laminar flow in the core and a turbulent flow on the periphery of the vortex. Research methods are entirely theoretical. Well-known theorems of theoretical mechanics, mathematical theory of field, and calculus of variations, etc. are used. The obtained solutions are compared with the existing corresponding analogs of non-compact flows. Conclusions. Using the methods of calculus of variation, it was possible to show the possibility of the formation of quasi-solid-like rotational motion in a boundary layer of an incompressible fluid. The very presence of viscosity, or rather its taking into account (boundary layer), indicates a possible transition of the flow from plane-parallel motion to the just-mentioned rotational one due to the Kelvin-Helmholtz instability. In addition, two new models of the Burgers-Rott vortex flow were obtained in this study. The first model uses the general solution obtained by Burgers, but this model corresponds to a combined vortex: although the velocity field is continuous, the vorticity field has a discontinuity - at the point of maximum velocity. It is proved that such an approach is quite possible: the equation of motion is satisfied everywhere, i.e., at every point in space, and the tangential stresses are continuous functions. Since the periphery of the Burgers-Rott vortex is an unstable flow, another model is proposed - with a laminar core and a turbulent periphery. Certainly, the motion of fluid parcels in the peripheral region is described by a velocity distribution other than that of Burgers. Finally, the possible use of known models of compact vortex flows to simulate the von Karman vortex street is considered, with an indication of the advantages of these models.

Under the hypothesis of electromagnetic waves propagation in inhomogeneous linear media with dielectric and magnetic anisotropy (in the absence of free charges and currents), analogies are presented between several discontinuity surfaces and wave-front propagation in these media. In particular, the "vortex sheet" model, borrowed from Fluid Mechanics is discussed. [ABSTRACT FROM AUTHOR]

A review of fundamental theoretical publications is presented, wherein the flow of an incompressible fluid in the cores of three-dimensional vortex sheets formed in the case of the separated flow of low-aspect-ratio wings are considered. Problems concerning an inviscid-fluid flow in the core of a conical vortex sheet and a vortex sheet vanishing from the edges of a parabolic wing, as well as the problem of a viscous-fluid flow in the core of a conical vortex structure, are considered. [ABSTRACT FROM AUTHOR]

A review of fundamental theoretical studies concerning the theory and application of the nonsteady-state analogy to an incompressible fluid flowing around slender bodies is presented. Problems related to the application of nonsteady-state analogy to wings, to the wake behind an elliptically loaded wing, as well as methods for numerical calculation of the evolution of vortex sheets and for determining the positions of lines of low separation from solid surfaces are considered taking into account viscous-inviscid interactions. The issues of nonuniqueness and asymmetry of solutions for the problems of a separated flow moving around slender bodies are discussed. [ABSTRACT FROM AUTHOR]

A review of fundamental theoretical studies concerning plane vortex flows in an incompressible fluid is presented. Problems connected with flow in the vicinity of the point of a vortex sheet vanishing from a solid surface, with self-similar flows of an ideal and viscous fluid, with flow in the cores of spiral vortex sheets, with the stability and diffusion of vortex flows, and with the development of a theory of boundary-layer separation from a solid surface are considered. [ABSTRACT FROM AUTHOR]

In this paper, we construct new, uniformly rotating solutions of the vortex sheet equation bifurcating from circles with constant vorticity amplitude. The proof is accomplished via a Lyapunov–Schmidt reduction and a second-order expansion of the reduced system. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'. [ABSTRACT FROM AUTHOR]

We continue the study of Confined Vortex Surfaces (CVS) that we introduced in the previous paper. We classify the solutions of the CVS equation and find the analytical formula for the velocity field for arbitrary background strain eigenvalues in the stable region. The vortex surface cross-section has the form of four symmetric hyperbolic sheets with a simple equation | y | | x | μ = const in each quadrant of the tube cross-section (x y plane). We use the dilute gas approximation for the vorticity structures in a turbulent flow, assuming their size is much smaller than the mean distance between them. We vindicate this assumption by the scaling laws for the surface shrinking to zero in the extreme turbulent limit. We introduce the Gaussian random background strain for each vortex surface as an accumulation of a large number of small random contributions coming from other surfaces far away. We compute this self-consistent background strain, relating the variance of the strain to the energy dissipation rate. We find a universal asymmetric distribution for energy dissipation. A new phenomenon is a probability distribution of the shape of the profile of the vortex tube in the x y plane. This phenomenon naturally leads to the "multifractal" scaling of the moments of velocity difference v ( r 1) − v ( r 2). More precisely, these moments have a nontrivial dependence of n , log Δ r , approximating power laws with effective index ζ (n , log Δ r). We derive some general formulas for the moments containing multidimensional integrals. The rough estimate of resulting moments shows the log–log derivative ζ (n , log Δ r) which is approximately linear in n and slowly depends on log Δ r. However, the value of effective index is wrong, which leads us to conclude that some other solution of the CVS equations must be found. We argue that the approximate phenomenological relations for these moments suggested in a recent paper by Sreenivasan and Yakhot are consistent with the CVS theory. We reinterpret their renormalization parameter α ≈ 0. 9 5 in the Bernoulli law p = − 1 2 α v 2 as a probability to find no vortex surface at a random point in space. [ABSTRACT FROM AUTHOR]

In this paper, we study the long-time evolutions of the single- and multi-mode Rayleigh–Taylor instability with surface tension in two dimensions by using the vortex sheet model. Applying a spectrally accurate numerical method, we investigate the effects of surface tension and density jump on the instability in various regimes of parameters. Complex phenomena of pinching, capillary waves, elongation, and roll-up appear at the interfaces. For a single-mode interface, surface tension retards the growths of bubble and spike. The effect of surface tension on the bubble and spike velocity is generally small but is large for a spike of an infinite density ratio. For multi-mode interfaces, we focus on an infinite density ratio. We show that bubbles grow with the scaling law h = α A g t 2 even in the presence of surface tension, while spikes follow the scaling law weakly. It is found that both the growth rates of bubbles and spikes decrease with surface tension and the growth rate of spikes decreases larger than that of bubbles. The growth rate of the bubble front is in agreements with results of previous numerical simulations and experiments. • The long-time evolution of Rayleigh–Taylor instability with surface tension is studied. • A spectrally accurate method based on the vortex sheet model is used for computations. • For a single-mode interface, surface tension retards the growths of bubble and spike. • For multi-mode interfaces, the growth rates of bubbles and spikes decrease with surface tension. [ABSTRACT FROM AUTHOR]

The equations of motion for an incompressible flow with helical symmetry (invariance under combined axial translation and rotation) can be expressed as nonlinear evolution laws for two scalars: vorticity and along-helix velocity. A metric term related to the pitch of the helix enters these equations, which reduce to two-dimensional and axisymmetric dynamics in appropriate limits. We take the vorticity and along-helix velocity component to be piecewise constant. In addition to this vortex patch, a vortex sheet develops when the along-helix velocity is nonzero. We obtain a contour dynamics formulation of the full nonlinear equations of motion, in which the motion of the boundary is computed in a Lagrangian fashion and the velocity field can be expressed as contour integrals, reducing the dimensionality of the computation. We investigate the stability properties of a circular vortex patch along the axis of the helix in the presence of a vortex sheet and along-helix velocity. A linear stability calculation shows that the system is stable when the initial vortex sheet is zero, but can be stable or unstable in the presence of a vortex sheet. Using contour dynamics, we examine the nonlinear evolution of the system, and show that nonlinear effects become important in unstable cases. [ABSTRACT FROM AUTHOR]

The motion of a two-dimensional buoyant vortex patch, i. e., a vortex patch with a uniform density different from the uniform density of the surrounding fluid, is analyzed in terms of evolution equations for the motion of its centroid, deformation of its boundary and the strength distribution of a vortex sheet which is essential to enforce pressure continuity across the boundary. The equations for the centroid are derived by a linear momentum analysis and that for the sheet strength distribution by applying Euler's equations on the boundary, while the boundary deformation is studied in the centroid-fixed frame. A complicated coupled set of equations is obtained which, to the best of our knowledge, has not been derived before. The evolution of the sheet strength distribution is obtained as an integral equation. The equations are also discussed in the limit of a patch of vanishing size or a buoyant point vortex. [ABSTRACT FROM AUTHOR]

We carried out the implementation of visualization techniques that aim at the interpretation of the complex turbulent structures in fluid flow. Turbulent structures help in understanding various phenomena of the flow field such as energy dissipation and drag reduction. However, few studies were found regarding vortex sheet structure identification for experimental data. In the current study, these sheet structures were visualized. Experimental data for the three-dimensional instantaneous velocity vector field was measured using the Tomographic PIV method. Two different kinds of flow settings, pipe flow and whirlpool flow, were measured and analyzed. For comparison of results, pipe measurements were recorded at varying Reynolds number and whirlpool flow were recorded with constant stirring and decaying turbulence. Q criterion analysis, which evaluates regions of higher vorticity than strain, was employed for visualization of vortex tubes in a time series flow field measurement. Vortex sheets were visualized using the Aij matrix analysis that utilizes second-order velocity gradient tensor and its eigenvalues for identifying regions with a high correlation and magnitudes of vorticity and strain. The relative position of the tube and sheet structures visualized using the two methods were reviewed for the experimental data of pipe flow and whirlpool flow. Comparison of the experimentally resolved structures with structures of DNS data was studied for pipe flow and a close resemblance was observed.

We revise the steady vortex surface theory following the recent finding of asymmetric vortex sheets (Migdal, 2021). These surfaces avoid the Kelvin–Helmholtz instability by adjusting their discontinuity and shape. The vorticity collapses to the sheet only in an exceptional case considered long ago by Burgers and Townsend, where it decays as a Gaussian on both sides of the sheet. In generic asymmetric vortex sheets (Shariff, 2021), vorticity leaks to one side or another, making such sheets inadequate for vortex sheet statistics and anomalous dissipation. We conjecture that the vorticity in a turbulent flow collapses on a special kind of surface (confined vortex surface or CVS), satisfying some equations involving the tangent components of the local strain tensor. The most important qualitative observation is that the inequality needed for this solution's stability breaks the Euler dynamics' time reversibility. We interpret this as dynamic irreversibility. We have also represented the enstrophy as a surface integral, conserved in the Navier–Stokes equation in the turbulent limit, with vortex stretching and viscous diffusion terms exactly canceling each other on the CVS surfaces. We have studied the CVS equations for the cylindrical vortex surface for an arbitrary constant background strain with two different eigenvalues. This equation reduces to a particular version of the stationary Birkhoff–Rott equation for the 2D flow with an extra nonanalytic term. We study some general properties of this equation and reduce its solution to a fixed point of a map on a sphere, guaranteed to exist by the Brouwer theorem. [ABSTRACT FROM AUTHOR]

We study steady vortex sheet solutions of the Navier–Stokes in the limit of vanishing viscosity at fixed energy flow. We refer to this as the turbulent limit. These steady flows correspond to a minimum of the Euler Hamiltonian as a functional of the tangent discontinuity of the local velocity parametrized as Δ v → t = ∇ → Γ. This observation means that the steady flow represents the low-temperature limit of the Gibbs distribution for vortex sheet dynamics with the normal displacement δ r ⊥ of the vortex sheet as a Hamiltonian coordinate and Γ as a conjugate momentum. An infinite number of Euler conservation laws lead to a degenerate vacuum of this system, which explains the complexity of turbulence statistics and provides the relevant degrees of freedom (random surfaces). The simplest example of a steady solution of the Navier–Stokes equation in the turbulent limit is a spherical vortex sheet whose flow outside is equivalent to a potential flow past a sphere, while the velocity is constant inside the sphere. Potential flow past other bodies provide other steady solutions. The new ingredient we add is a calculable gap in tangent velocity, leading to anomalous dissipation. This family of steady solutions provides an example of the Euler instanton advocated in our recent work, which is supposed to be responsible for the dissipation of the Navier–Stokes equation in the turbulent limit. We further conclude that one can obtain turbulent statistics from the Gibbs statistics of vortex sheets by adding Lagrange multipliers for the conserved volume inside closed surfaces, the rate of energy pumping, and energy dissipation. The effective temperature in our Gibbs distribution goes to zero as Re − 1 3 with Reynolds number Re ∼ ν − 6 5 in the turbulent limit. The Gibbs statistics in this limit reduces to the solvable string theory in two dimensions (so-called c = 1 critical matrix model). This opens the way for nonperturbative calculations in the Vortex Sheet Turbulence, some of which we report here. [ABSTRACT FROM AUTHOR]

We propose a spectral viscosity method to approximate the two-dimensional Euler equations with rough initial data and prove that the method converges to a weak solution for a large class of initial data, including when the initial vorticity is in the so-called Delort class, i.e., it is a sum of a signed measure and an integrable function. This provides the first convergence proof for a numerical method approximating the Euler equations with such rough initial data and closes the gap between the available existence theory and rigorous convergence results for numerical methods. We also present numerical experiments, including computations of vortex sheets and confined eddies, to illustrate the proposed method. [ABSTRACT FROM AUTHOR]

We consider supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients, Morando et al. recently derived a pseudo-differential equation that describes the time evolution of the discontinuity front of the vortex sheet. In agreement with the classical stability analysis, the problem is weakly stable if | [ v ⋅ τ ] | > 2 2 c , and the well-posedness holds in standard weighted Sobolev spaces. Our aim in this paper is to improve this result, by showing the existence in functional spaces with additional weighted anisotropic regularity in the frequency space. [ABSTRACT FROM AUTHOR]

An original algorithm is developed for vortex methods of computational fluid dynamics for determining the intensity of the vortex sheet on the surface of a body in the flow of an incompressible medium. Unlike the common in the vortex methods approach to satisfying the no-slip boundary condition on a streamlined surface, which is based on ensuring that the normal velocity component of the medium is zero, the proposed procedure is based on a mathematically equivalent condition of equality to zero of the tangent velocity component on the body surface. The unknown intensity of the vortex sheet is assumed to be piecewise constant on triangular panels that approximate the surface of the body. The resulting integral equation is approximated by a system of linear algebraic equations, which dimension is twice the number of panels. The coefficients of the system matrix are expressed through double integrals over the panels. An algorithm is proposed for calculating these integrals for the case of neighboring panels, when these integrals are improper. An additive singularity exclusion is performed and analytical expressions for the integrals of them are obtained. The smooth parts of integrands are integrated numerically using Gaussian quadrature formulae. The proposed algorithm makes it possible to improve significantly the accuracy of the vortex sheet intensity reconstruction when flow simulating around complex-shaped bodies by using vortex methods for arbitrary triangular surface meshes, including essentially non-uniform and having cells with high aspect ratio. [ABSTRACT FROM AUTHOR]

An inviscid vortex shedding model for separated vortices from a solid body is studied. The model describes the separated vortices by vortex sheets and the attached flow via conformal mapping. We develop a computational model to simulate the vortex shedding of a moving body, with varying angle. An unsteady Kutta condition is imposed on the edges of the plate to determine the edge circulations and velocities. The force on the plate is obtained by integrating the unsteady Blasius equation. We apply the model to two representative cases of an accelerated plate, with impulsive start and uniform acceleration, and investigate the dynamics for large angles of attack. For both cases, the vortex force is dominant in the lift over times. The lift coefficients are initially high and decrease in four chord lengths of displacement, in general. For large angles of attack, the appearance of a peak of lift at an early time depends on the power-law velocity, which differs from the behavior for small angles of attack. The lift and drag from the model are in agreement with the Navier–Stokes simulation and experiment for moderate Reynolds numbers. We also demonstrate the vortex shedding of hovering and flapping plates. In the hovering motion, the large increase in lift at the early backward translation is due to the combined effect of the vortex force and added mass force. In the flapping plate, our model provides an improvement in the prediction for the induced force than other shedding models. [ABSTRACT FROM AUTHOR]

We are concerned with supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients we derive an evolution equation for the discontinuity front of the vortex sheet. This is a pseudo‐differential equation of order two. In agreement with the classical stability analysis, if the Mach number M satisfies M<2, the symbol is elliptic and the problem is ill‐posed. On the contrary, if M>2 then the problem is weakly stable, and we are able to derive a wave‐type a priori energy estimate for the solution, with no loss of regularity with respect to the data. Then we prove the well‐posedness of the problem, by showing the existence of the solution in weighted Sobolev spaces. [ABSTRACT FROM AUTHOR]

An interesting physical phenomenon was recently observed when a fresh-water basin is covered by a thin ice film that has properties similar to the property of a rubber membrane. Surface waves can be generated under the action of wind on the air–water interface that contains an ice film. The modulation property of hydro-elastic waves (HEWs) in deep water covered by thin ice film blown by the wind with a uniform vertical profile is studied here in terms of the airflow velocity versus wavenumber. The modulation instability of HEWs is studied through the analysis of coefficients of the nonlinear Schrödinger (NLS) equation with the help of the Lighthill criterion. The NLS equation is derived using the multiple scale method in the presence of airflow. It is demonstrated that the potentially unstable hydro-elastic waves with negative energy appear for relatively small wind speeds, whereas the Kelvin–Helmholtz instability arises when the wind speed becomes fairly strong. Estimates of parameters of modulated waves for the typical conditions are given.