Geophysical research has focused on flows, such as ocean currents, as two dimensional. Two dimensional point or blob vortex models have the advantage of having a Hamiltonian, whereas 3D vortex filament or tube systems do not necessarily have one, although they do have action functionals. On the other hand, certain classes of 3D vortex models called nearly parallel vortex filament models do have a Hamiltonian and are more accurate descriptions of geophysical and atmospheric flows than purely 2D models, especially at smaller scales. In these quasi-2D'' models we replace 2D point vortices with vortex filaments that are very straight and nearly parallel but have Brownian variations along their lengths due to local self-induction. When very straight, quasi-2D filaments are expected to have virtually the same planar density distributions as 2D models. An open problem is when quasi-2D model statistics behave differently than those of the related 2D system and how this difference is manifested. In this paper we study the nearly parallel vortex filament model of Klein, Majda, Damodaran in statistical equilibrium. We are able to obtain a free-energy functional for the system in a non-extensive thermodynamic limit that is a function of the mean square vortex position @math and solve for @math . Such an explicit formula has never been obtained for a non-2D model. We compare the results of our formula to a 2-D formula of Lim:2005 and show qualitatively different behavior even when we disallow vortex braiding. We further confirm our results using Path Integral Monte Carlo (Ceperley (1995)) permutations and that the Klein, Majda, Damodaran model's asymptotic assumptions for parameters where these deviations occur.
@cite_12 has done some excellent simulations of vortex tangles in He-4 with rotation, boundary walls, and vortex reconnections to study disorder in rotating superfluid turbulence. Because vortex tangles are extremely curved, they applied the full Biot-Savart law to calculate the motion of the filaments in time. Their study did not include any sort of comparison to 2-D models because for most of the simulation vortices were far too tangled. The inclusion of rigid boundary walls, although correct for the study of He-4, also makes the results only tangentially applicable to the KMD system we use.
@cite_12
2586908737 2010371781 2129152507 1966911335
Turbulent vortices in smoke flows are crucial for a visually interesting appearance. Unfortunately, it is challenging to efficiently simulate these appealing effects in the framework of vortex filament methods. The vortex filaments in grids scheme allows to efficiently generate turbulent smoke with macroscopic vortical structures, but suffers from the projection-related dissipation, and thus the small-scale vortical structures under grid resolution are hard to capture. In addition, this scheme cannot be applied in wall-bounded turbulent smoke simulation, which requires efficiently handling smoke-obstacle interaction and creating vorticity at the obstacle boundary. To tackle above issues, we propose an effective filament-mesh particle-particle (FMPP) method for fast wall-bounded turbulent smoke simulation with ample details. The Filament-Mesh component approximates the smooth long-range interactions by splatting vortex filaments on grid, solving the Poisson problem with a fast solver, and then interpolating back to smoke particles. The Particle-Particle component introduces smoothed particle hydrodynamics (SPH) turbulence model for particles in the same grid, where interactions between particles cannot be properly captured under grid resolution. Then, we sample the surface of obstacles with boundary particles, allowing the interaction between smoke and obstacle being treated as pressure forces in SPH. Besides, the vortex formation region is defined at the back of obstacles, providing smoke particles flowing by the separation particles with a vorticity force to simulate the subsequent vortex shedding phenomenon. The proposed approach can synthesize the lost small-scale vortical structures and also achieve the smoke-obstacle interaction with vortex shedding at obstacle boundaries in a lightweight manner. The experimental results demonstrate that our FMPP method can achieve more appealing visual effects than vortex filaments in grids scheme by efficiently simulating more vivid thin turbulent features. We show that the stochastic differential equation (SDE) model for the merger of two identical two-dimensional vortices proposed by Agullo and Verga [“Exact two vortices solution of Navier–Stokes equation,” Phys. Rev. Lett. 78, 2361 (1997)] is a special case of a more general class of SDE models for N interacting vortex filaments. These toy models include vorticity diffusion via a white noise forcing of the inviscid equations, and thus extend inviscid models to include core dynamics and topology change (e.g., merger in two dimensions and vortex reconnection in three dimensions). We demonstrate that although the N=2 two-dimensional model is qualitatively and quantitatively incorrect, it can be dramatically improved by accounting for self-advection. We then extend the two-dimensional SDE model to three dimensions using the semi-inviscid asymptotic approximation of [“Simplified equations for the interactions of nearly parallel vortex filaments,” J. Fluid Mech. 288, 201 (1995)] for nearly parallel... Abstract This paper reports large-scale direct numerical simulations of homogeneous-isotropic fluid turbulence, achieving sustained performance of 1.08 petaflop s on gpu hardware using single precision. The simulations use a vortex particle method to solve the Navier–Stokes equations, with a highly parallel fast multipole method ( fmm ) as numerical engine, and match the current record in mesh size for this application, a cube of 4096 3 computational points solved with a spectral method. The standard numerical approach used in this field is the pseudo-spectral method, relying on the fft algorithm as the numerical engine. The particle-based simulations presented in this paper quantitatively match the kinetic energy spectrum obtained with a pseudo-spectral method, using a trusted code. In terms of parallel performance, weak scaling results show the fmm -based vortex method achieving 74 parallel efficiency on 4096 processes (one gpu per mpi process, 3 gpu s per node of the tsubame -2.0 system). The fft -based spectral method is able to achieve just 14 parallel efficiency on the same number of mpi processes (using only cpu cores), due to the all-to-all communication pattern of the fft algorithm. The calculation time for one time step was 108 s for the vortex method and 154 s for the spectral method, under these conditions. Computing with 69 billion particles, this work exceeds by an order of magnitude the largest vortex-method calculations to date. New analytical solutions are presented for steadily translating von Karman vortex streets made up of two infinite rows of hollow vortices. First, the solution for a single row of hollow vortices due to ["Structure of a linear array of hollow vortices of finite cross-section," J. Fluid Mech. 74, 469 (1976)] is rederived, in a modified form, and using a new mathematical approach. This approach is then generalized to find relative equilibria for both unstaggered and staggered double hollow vortex streets. The method employs a combination of free streamline theory and conformal mapping ideas. The staggered hollow vortex streets are compared with analogous numerical solutions for double streets of vortex patches due to Saffman and Schatzman [“Properties of a vortex street of finite vortices,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 2, 285 (1981)] and several common features are found. In particular, within the two different inviscid vortex models, the same street aspect ratio of approxi...