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Register a new userDevelopment of high vorticity structures and geometrical properties of the vortex line representation
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The incompressible three-dimensional Euler equations develop very thin pancake-like regions of increasing vorticity. These regions evolve with the scaling $omega_{max}simell^{-2/3}$ between the vorticity maximum and the pancake thickness, as was observed in the recent numerical experiments [D.S. Agafontsev et al, Phys. Fluids 27, 085102 (2015)]. We study the process of pancakes development in terms of the vortex line representation (VLR), which represents a partial integration of the Euler equations with respect to conservation of the Cauchy invariants and describes compressible dynamics of continuously distributed vortex lines. We present, for the first time, the numerical simulations of the VLR equations with high accuracy, which we perform in adaptive anisotropic grids of up to $1536^3$ nodes. With these simulations, we show that the vorticity growth is connected with the compressibility of the vortex lines and find geometric properties responsible for the observed scaling $omega_{max}simell^{-2/3}$.
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The incompressible three-dimensional ideal flows develop very thin pancake-like regions of increasing vorticity. These regions evolve with the scaling $omega_{max}(t)proptoell(t)^{-2/3}$ between the vorticity maximum and pancake thickness, and provide the leading contribution to the energy spectrum, where the gradual formation of the Kolmogorov interval $E_{k}propto k^{-5/3}$ is observed for some initial flows [Agafontsev et. al, Phys. Fluids 27, 085102 (2015)]. With the massive numerical simulations, in the present paper we study the influence of initial conditions on the processes of pancake formation and the Kolmogorov energy spectrum development.
Issues relevant to the flow chirality and structure are focused, while the new theoretical results, including even a distinctive theory, are introduced. However, it is hope that the presentation, with a low starting point but a steep rise, is appropriate for a broader spectrum of audiences ranging from students to researchers, thus illustrations of differential forms and relevant basic topological concepts are also offered, followed by the demonstration with formulation of differential forms of the classical Navier-Stokes flow theory and the discussions of recent studies in fundamental fluid mechanics and turbulence.
We discuss the statistical properties of a single vortex line in a perfect fluid. The partition function is calculated up to the end in the thin vortex approximation. It turns out that corresponding theory is renormalizable, and the renormalization law for the core size of the vortex is found. The direction of renormalization group flow makes the thin vortex approximation to be valid for the interest cases and this result does not depend on the choice of infrared regularization. The expressions for some gauge-invariant correlators are obtained to demonstrate the developed formalism at work.
Shear layer instability at the free surface of a water jet is studied. The accompanying video shows experimental data recorded using measurement methods such as Planar Laser Induced Fluorescence (PLIF) and Particle Image Velocity (PIV). These results reveal the mechanisms leading to the formation of capillary waves on the surface due to the roll-up of the shear layer. These capillary waves eventually collide to each other, injecting vorticity in the bulk of the flow. Shear layer and injected vorticity interact to form a counter rotating vortex pair that moves down to the flow.
In rotating Rayleigh-Benard convection, columnar vortices advect horizontally in a stochastic manner. When the centrifugal buoyancy is present the vortices exhibit radial motions that can be explained through a Langevin-type stochastic model. Surprisingly, anomalous outward motion of cyclones is observed in a centrifugation-dominant flow regime, which is contrary to the well-known centrifugal effect. We interpret this phenomenon as a symmetry-breaking of both the population and vorticity magnitude of the vortices brought about by the centrifugal buoyancy. Consequently, the cyclones submit to the collective vortex motion dominated by the strong anticyclones. Our study provides new understanding of vortex motions that are widely present in many natural systems.
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