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Register a new userErgodicity and spectral cascades in point vortex flows on the sphere
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We present results for the equilibrium statistics and dynamic evolution of moderately large ($n = {mathcal{O}}(10^2 - 10^3)$) numbers of interacting point vortices on the unit sphere under the constraint of zero mean angular momentum. We consider a binary gas consisting of equal numbers of vortices with positive and negative circulations. When the circulations are chosen to be proportional to $1/sqrt{n}$, the energy probability distribution function, $p(E)$, converges rapidly with $n$ to a function that has a single maximum, corresponding to a maximum in entropy. Ensemble-averaged wavenumber spectra of the nonsingular velocity field induced by the vortices exhibit the expected $k^{-1}$ behavior at small scales for all energies. The spectra at the largest scales vary continuously with the inverse temperature $beta$ of the system and show a transition from positively sloped to negatively sloped as $beta$ becomes negative. The dynamics are ergodic and, regardless of the initial configuration of the vortices, statistical measures simply relax towards microcanonical ensemble measures at all observed energies. As such, the direction of any cascade process measured by the evolution of the kinetic energy spectrum depends only on the relative differences between the initial spectrum and the ensemble mean spectrum at that energy; not on the energy, or temperature, of the system.
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At high Reynolds number, the interaction between two vortex tubes leads to intense velocity gradients, which are at the heart of fluid turbulence. This vorticity amplification comes about through two different instability mechanisms of the initial vortex tubes, assumed anti-parallel and with a mirror plane of symmetry. At moderate Reynolds number, the tubes destabilize via a Crow instability, with the nonlinear development leading to strong flattening of the cores into thin sheets. These sheets then break down into filaments which can repeat the process. At higher Reynolds number, the instability proceeds via the elliptical instability, producing vortex tubes that are perpendicular to the original tube directions. In this work, we demonstrate that these same transition between Crow and Elliptical instability occurs at moderate Reynolds number when we vary the initial angle $beta$ between two straight vortex tubes. We demonstrate that when the angle between the two tubes is close to $pi/2$, the interaction between tubes leads to the formation of thin vortex sheets. The subsequent breakdown of these sheets involves a twisting of the paired sheets, followed by the appearance of a localized cloud of small scale vortex structures. At smaller values of the angle $beta$ between the two tubes, the breakdown mechanism changes to an elliptic cascade-like mechanism. Whereas the interaction of two vortices depends on the initial condition, the rapid formation of fine-scales vortex structures appears to be a robust feature, possibly universal at very high Reynolds numbers.
We discuss the possibility of dual local and non-local cascades in a 3D turbulent Beltrami flow, with inverse energy cascade and direct helicity cascade, by analogy with 2D turbulence. We discuss the corresponding energy spectrum in both local and non-local case. Comparison with a high Reynolds number turbulent von Karman flow is provided and discussed.
Velocity gradient is the basis of many vortex recognition methods, such as Q criterion, $Delta$ criterion, $lambda_{2}$ criterion, $lambda_{ci}$ criterion and $Omega$ criterion, etc.. Except the $lambda_{ci}$ criterion, all these criterions recognize vortices by designing various invariants, based on the Helmholtz decomposition that decomposes velocity gradient into strain rate and spin. In recent years, the intuition of no vortex in straight flows has promoted people to analyze the vortex state directly from the velocity gradient, in which vortex can be distinguished from the situation that the velocity gradient has couple complex eigenvalues. A specious viewpoint to adopt the simple shear as an independent flow mode was emphasized by many authors, among them, Kolar proposed the triple decomposition of motion by extracting a so-called effective pure shearing motion; Li et al. introduced the so-called quaternion decomposition of velocity gradient and proposed the concept of eigen rotation; Liu et al. further mined the characteristic information of velocity gradient and put forward an effective algorithm of Liutex, and then developed the vortex recognition method. However, there is another explanation for the increasingly clear representation of velocity gradient, that is the local streamline pattern based on critical-point theory. In this paper, the tensorial expressions of the right/left real Schur forms of velocity gradient are clarified from the characteristic problem of velocity gradient. The relations between the involved parameters are derived and numerically verified. Comparing with the geometrical features of local streamline pattern, we confirm that the parameters in the right eigen-representation based on the right real Schur form of velocity gradient have good meanings to reveal the local streamline pattern. Some illustrative examples from the DNS data are presented.
Viscous heating can play an important role in the dynamics of fluids with strongly temperature-dependent viscosities because of the coupling between the energy and momentum equations. The heat generated by viscous friction produces a local temperature increase near the tube walls with a consequent decrease of the viscosity and a strong stratification in the viscosity profile. The problem of viscous heating in fluids was investigated and reviewed by Costa & Macedonio (2003) because of its important implications in the study of magma flows. Because of the strong coupling between viscosity and temperature, the temperature rise due to the viscous heating may trigger instabilities in the velocity field, which cannot be predicted by a simple isothermal Newtonian model. When viscous heating produces a pronounced peak in the temperature profile near the walls, a triggering of instabilities and a transition to secondary flows can occur because of the stratification in the viscosity profile. In this paper we focus on the thermal and mechanical effects caused by viscous heating. We will present the linear stability equations and we will show, as in certain regimes, these effects can trigger and sustain a particular class of secondary rotational flows which appear organised in coherent structures similar to roller vortices. This phenomenon can play a very important role in the dynamics of magma flows in conduits and lava flows in channels and, to our knowledge, it is the first time that it has been investigated by a direct numerical simulation.
The Tayler instability is a kink-type flow instability which occurs when the electrical current through a conducting fluid exceeds a certain critical value. Originally studied in the astrophysical context, the instability was recently shown to be also a limiting factor for the upward scalability of liquid metal batteries. In this paper, we continue our efforts to simulate this instability for liquid metals within the framework of an integro-differential equation approach. The original solver is enhanced by multi-domain support with Dirichlet-Neumann partitioning for the static boundaries. Particular focus is laid on the detailed influence of the axial electrical boundary conditions on the characteristic features of the Tayler instability, and, secondly, on the occurrence of electro-vortex flows and their relevance for liquid metal batteries.
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