Vorticity equation

Résumé

The vorticity equation of fluid dynamics describes the evolution of the vorticity ω of a particle of a fluid as it moves with its flow; that is, the local rotation of the fluid (in terms of vector calculus this is the curl of the flow velocity). The governing equation is:where ''D''/''Dt'' is the material derivative operator, u is the flow velocity, ρ is the local fluid density, p is the local pressure, τ is the viscous stress tensor and B represents the sum of the external body forces. The first source term on the right hand side represents vortex stretching. The equation is valid in the absence of any concentrated torques and line forces for a compressible, Newtonian fluid. In the case of incompressible flow (i.e., low Mach number) and isotropic fluids, with conservative body forces, the equation simplifies to the vorticity transport equation: :\frac{D\boldsymbol\omega}{Dt} = \left(\boldsymbol{\omega} \cdot \nabla\right) \m

Source officielle

https://en.wikipedia.org/wiki/Vorticity_equation

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Advancing SOL simulations: avoiding the Boussinesq approximation and coupling closed and open magnetic flux surfaces

Baptiste Jimmy Frei, Federico David Halpern, Jorge Ari Morales Mena, Félix Benedito Clément Musil, Paola Paruta, Paolo Ricci, Fabio Riva, Morgan Siffert, Christoph Wersal

Plasma turbulence in the tokamak scrape-off layer (SOL) region, where magnetic field lines intersect the reactor inner walls, determines the heat load on the limiter or divertor targets. This is one of the most crucial issues on the way towards a fusion reactor. To tackle this problem we have improved the turbulent model in the GBS code, a new formulation of the vorticity equation that allow us to relax the Boussinesq approximation is implemented. The energy conservation properties of the new system of equations are evaluated. Also results of turbulent simulations in the SOL with and without the Boussinesq approximation are compared. In addition turbulent simulations across the last closed flux surface taking into account a cold and a hot ion regime are shown. These last simulations show a pressure gradient increase at the separatrix in the hot ion case at low resistivity.

2016

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Large eddy simulation of the tip-leakage cavitating flow with an insight on how cavitation influences vorticity and turbulence

Huaiyu Cheng, Mohamed Farhat

Cavitation within a tip leakage flow remains a challenging issue in a variety of axial hydraulic machines. It is still not possible nowadays to predict cavitation occurrence in such a flow with acceptable accuracy. In the present study, we have carried out numerical simulations of a tip leakage cavitating flow, generated by a straight NACA0009 hydrofoil. We have used the LES method combined with the Schnerr-Sauer cavitation model. The numerical results agree well with experimental data. The evolution of the tip leakage cavitating flow, involving tip-leakage vortex (TLV), tip-separation vortex (TSV) and induced vortex (IV), is analyzed from Eulerian and Lagrangian viewpoints. The results show that the spatial evolution of the tip leakage cavitating flow can be divided into three stages: Stage I, Independent development of the TLV and TSV: Stage II, Fusion of the TLV and TSV; and Stage III, Development of the IV and dissipation of the TLV. The Lagrangian coherent structures (LCSs) obtained from the numerical results indicate that the TLV cavitation significantly influence the local flow patterns. The vorticity transport equation was then used to further analyze the influence of the cavitation on the vortices. The results demonstrate that the stretching term dominates the TLV evolution and the dilatation term is responsible for the vorticity reduction inside the TLV cavity. The results also show how the cavitation influences the local turbulence and that the transport term in the turbulent kinetic energy equation influences the turbulence distribution near the TLV cavity. (C) 2019 Elsevier Inc. All rights reserved.

ELSEVIER SCIENCE INC2020

Chargement

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In the tokamak scrape-off layer (SOL) the magnetic field lines are open, channeling particles and heat onto plasma facing components, and constraining their lifetime. Therefore, safe operation of future fusion reactors requires an understanding of SOL plasma dynamics. Recently, we have gained deep insights into the tokamak SOL dynamics by means of massively-parallel fluid electromagnetic turbulence simulations carried out with the Global Braginskii Solver (GBS) code. GBS is currently capable of performing full-size SOL simulations of medium-size tokamaks such as TCV or Alcator C-Mod. In the present paper, we emphasize recent numerical developments aimed towards (a) more realistic description of the plasma geometry and (b) simulating larger, reactor-class machines. First, we address the numerical implementation of parallel advection and diffusion operators in finite difference representation. This is a crucial aspect of our computation, as the cost of the simulation can be drastically reduced if the parallel dynamics are discretized adequately taking advantage of the strong anisotropy of the turbulent modes that are aligned to the magnetic field lines. Second, we address the development and implementation of a matrix-free, parallel multigrid solver for the Poisson operator in the vorticity equation. Using this new solver, it is now possible to further parallelize GBS without breaking parallel scalability, an important step towards the realm of 10^4 CPUs. Finally, we summarize the understanding of SOL turbulence obtained through GBS simulations - for instance, the mechanisms regulating the turbulence level and therefore the SOL width, the turbulent regimes, and the mechanisms driving plasma rotation in this region. The simulated non-linear dynamics have been compared with analytical estimates, which have highlighted the key physics mechanisms at play in the SOL, and with experimental measurements taken in a number of tokamak worldwide, showing good agreement.

2014

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2014