Quantized vortices in pionic superfluid

The interplay between classical vorticity being the main undisputed source of polarization in heavy-ion collisions (HIC) and quantized vortices is considered. The vortex tubes emerging in the rotating pionic (super) fluid polarize the baryons in their cores and explain the emerging global polarization. The appearance of vortices in the region separating participants and spectators in non-central HIC is similar to that for sliding layers of liquid helium. From the other side, it is also the region where the classical vorticity was earlier found to be large forming the vortex sheets. The formation of tubes manifests a threshold at certain critical vorticity implying the vanishing polarization at lower energies. For central HIC the compact jet-like flows may lead to formation of vortex rings related to local polarization. The P-odd momentum correlations for their experimental investigation are suggested. The role of shear and viscosity in the emergence of polarization is discussed.

Complex singularity analysis for vortex layer flows

We study the evolution of a 2D vortex layer at high Reynolds number. Vortex layer flows are characterized by intense vorticity concentrated around a curve. In addition to their intrinsic interest, vortex layers are relevant configurations because they are regularizations of vortex sheets. In this paper, we consider vortex layers whose thickness is proportional to the square-root of the viscosity. We investigate the typical roll-up process, showing that crucial phases in the initial flow evolution are the formation of stagnation points and recirculation regions. Stretching and folding characterizes the following stage of the dynamics, and we relate these events to the growth of the palinstrophy. The formation of an inner vorticity core, with vorticity intensity growing to infinity for larger Reynolds number, is the final phase of the dynamics. We display the inner core's self-similar structure, with the scale factor depending on the Reynolds number. We reveal the presence of complex singularities in the solutions of Navier–Stokes equations; these singularities approach the real axis with increasing Reynolds number. The comparison between these singularities and the Birkhoff–Rott singularity seems to suggest that vortex layers, in the limit $Re\rightarrow \infty$ , behave differently from vortex sheets.

Confined vortex surface and irreversibility. 1. Properties of exact solution

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.

Efficient computation of coordinate-free models of flame fronts

We present an efficient, accurate computational method for a coordinate-free model of flame front propagation of Frankel and Sivashinsky. This model allows for overturned flames fronts, in contrast to weakly nonlinear models such as the Kuramoto–Sivashinsky equation. The numerical procedure adapts the method of Hou, Lowengrub and Shelley, derived for vortex sheets, to this model. The result is a nonstiff, highly accurate solver which can handle fully nonlinear, overturned interfaces, with similar computational expense to methods for weakly nonlinear models. We apply this solver both to simulate overturned flame fronts and to compare the accuracy of Kuramoto–Sivashinsky and coordinate-free models in the appropriate limit. doi:10.1017/S1446181121000079

Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $$\Omega $$ Ω , such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.

EFFICIENT COMPUTATION OF COORDINATE-FREE MODELS OF FLAME FRONTS

We present an efficient, accurate computational method for a coordinate-free model of flame front propagation of Frankel and Sivashinsky. This model allows for overturned flames fronts, in contrast to weakly nonlinear models such as the Kuramoto–Sivashinsky equation. The numerical procedure adapts the method of Hou, Lowengrub and Shelley, derived for vortex sheets, to this model. The result is a nonstiff, highly accurate solver which can handle fully nonlinear, overturned interfaces, with similar computational expense to methods for weakly nonlinear models. We apply this solver both to simulate overturned flame fronts and to compare the accuracy of Kuramoto–Sivashinsky and coordinate-free models in the appropriate limit.

Collapse of n Point Vortices, Formation of the Vortex Sheets and Transport of Passive Markers

In this paper, the motion of the n-vortex system as it collapses to a point in finite time is studied. The motion of vortices is described by the set of ordinary differential equations that we are able to solve analytically. The explicit formula for the solution demands the initial location of collapsing vortices. To find the collapsing locations of vortices, the algebraic, nonlinear system of equations was built. The solution of that algebraic system was obtained using Newton’s procedure. A good initial iterate needs to be provided to succeed in the application of Newton’s procedure. An unconstrained Leverber–Marquart optimization procedure was used to find such a good initial iterate. The numerical studies were conducted, and numerical evidence was presented that if in a collapsing system n=50 point vortices include a few vortices with much greater intensities than the others in the set, the vortices with weaker intensities organize themselves onto the vortex sheet. The collapsing locations depend on the value of the Hamiltonian. By changing the Hamiltonian values in a specific interval, the collapsing curves can be obtained. All points on the collapse curves with the same Hamiltonian value represent one collapsing system of vortices. To show the properties of vortex sheets created by vortices, the passive tracers were used. Advection of tracers by the velocity induced by vortices was calculated by solving the proper differential equations. The vortex sheets are an impermeable barrier to inward and outward fluxes of tracers. Arising vortex structures are able to transport the passive tracers. In this paper, several examples showing the diversity of collapsing structures with the vortex sheet are presented. The collapsing phenomenon of many vortices, their ability to self organize and the transportation of the passive tracers are novelties in the context of point vortex dynamics.

An impulse-based derivation of the Kutta–Joukowsky equation for wind turbine thrust

Using the concept of impulse in control volume analysis, we derive general expressions for wind turbine thrust in a constant, spatially uniform wind. The absence of pressure in the impulse equations allows for their application in the near wake, where the flow is assumed to be steady in the frame of reference rotating with the blades. The assumption of circumferential uniformity in the near wake – as applies when the number of blades or the tip speed ratio tends to infinity – is needed to reduce these general expressions to the Kutta–Joukowsky (KJ) equation for blade-element thrust. The present derivation improves upon the classical derivation based on the Bernoulli equation by allowing the flow to be rotational in the near wake. The present derivation also yields intermediate expressions for thrust that are valid for a finite number of blades and trailing vortex sheets of finite thickness. For the circumferentially uniform case, our analysis suggests that the magnitudes of the radial velocity and the axial induction factor must be equal somewhere on the plane containing the rotor, and we cite previous studies that show this to occur near the rotor tip across a wide range of thrust coefficients. The derivation reveals one further complication; when deriving the KJ equations using annular control volumes, the existence of vorticity on the lateral control surfaces may cause the local blade loading to differ from the KJ equation, but the magnitude of these deviations is not explored. This complication is not visible to the classical derivation due to its neglect of vorticity.