Magnetic field based approximation for azimuthal dependent magnetic permeability

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Magnetic permeability jumps in r and z

We end our description of SFEMaNS possibilities with the last development implemented before this PhD period: the implementation of magnetic permeability jumps in r and z directions. This new method, implemented during the thesis period of F. Luddens [74], allows jumps in the electrical conductivity and magnetic permeability between the fluid domain c;f and the different conducting solid domains i c;s introduced in section 2.1.
The main difficulties of such problems are to satisfy continuity conditions across interfaces between c;f , i c;s, v and to impose a zero divergence on the induction field B = H. Indeed by denoting the interface between conducting domains and the interface between insulating domain v with conducting domains, approximations need to satisfy.

Extension to non axisymmetric geometry

In this section we present a method, called pseudo-penalization and introduced by Pasquetti et al. [89], which we implemented in SFEMaNS to consider non axisymmetric geometry. Firstly we describe the method and give details on its use with prediction-correction scheme for the Navier-Stokes equations. Then we present various numerical tests involving manufactured solutions or physical problems so we can attest of the correct behavior of the method and enhance some of its properties.

Pseudo-penalization method and prediction-correction scheme forthe Navier-Stokes equations

The goal of the following is to describe a technique which allows us to take into account a non axisymmetric domain that we split into a fluid domain, denoted by fluid , and a solid domain, denoted by obs. While we would like to approximate the solutions of the Navier-Stokes equations (1.3.5) in fluid, we also want to enforce the velocity field to be zero in the solid domain obs that represents an obstacle. To do so we plan to use a pseudo-penalization method that is described in the following for prediction-correction scheme. Eventually we give details on its implementation in the SFEMaNS code, while extending the method to solid obstacles with non zero velocity uobs. In the following the time step is denoted by , a function at time tn = n is denoted fn.

Extension to MHD problems with variable fluid and solid properties

During this PhD period we focused our investigations on three subjects: precession, VKS experiment and multiphase flow problems. While the precession only presents computational cost difficulties due to the Reynolds numbers involved, the other two problems involve the implementation of new approximation methods in the code. While the pseudopenalization method described in section 2.4 extends the range of SFEMaNS code to hydrodynamic problems with non axisymmetric geometry, it remains to take into account magnetic problems with a given time and (r; ; z) dependent magnetic permeability and hydrodynamic problems with variable density and viscosity.
In order to approximate such problems we have the choice to approximate the Maxwell equations either with the magnetic field H or the induction field B = H with the magnetic permeability. In the same way, the Navier-Stokes equations can either be approximated with the velocity field u or the momentum m = u with the density. A first study, done with D. Castanon-Quiroz during the first part of a one year stay at Texas A&M University (College Station, Texas) thanks to an invitation of J.-L. Guermond, draws us to focus on the following simplified equations: @t(u) 􀀀 rru = 0.

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Table of contents :

1 Introduction 
1.1 Context and motivations
1.2 Thesis outline
1.3 Magnetohydrodynamic equations
1.3.1 Navier-Stokes Equations
1.3.2 Maxwell Equations
1.3.3 Magnetohydrodynamic Equations
2 SFEMaNS MHD-code 
2.1 Framework
2.2 Numerical approximation
2.2.1 Fourier discretization
2.2.2 Finite Element representation
2.3 SFEMaNS possibilities
2.3.1 Parallelization
2.3.2 Heat Equation
2.3.3 Magnetic permeability jumps in r and z
2.4 Extension to non axisymmetric geometry
2.4.1 Pseudo-penalization method and prediction-correction scheme for the Navier-Stokes equations
2.4.2 Numerical test with manufactured solutions
2.4.3 Flow past a sphere and drag coefficient
2.5 Extension to MHD problems with variable fluid and solid properties
2.5.1 Magnetic field based approximation for azimuthal dependent magnetic permeability
2.5.2 Momentum based approximation for multiphase flow problems
2.6 Outlook
3 Nonlinear stabilization method: entropy viscosity 
3.1 Context and method
3.1.1 On the need of models
3.1.2 Large Eddy Simulation models
3.1.3 Entropy viscosity as LES method
3.2 Entropy viscosity and SFEMaNS code
3.2.1 Numerical Implementation
3.2.2 Numerical tests
3.2.3 Outlook
4 Large Eddy Simulation with entropy viscosity 
4.1 Hydrodynamic study of a Von Kármán Sodium set-up
4.1.1 Experimental set-up
4.1.2 Numerical approximation
4.1.3 Hydrodynamic regimes for Re 2500
4.1.4 Numerical results with entropy viscosity method
4.1.5 Conclusion
4.2 Two spinning ways for precession dynamo
4.2.1 Introduction
4.2.2 Numerical settings
4.2.3 Hydrodynamic study
4.2.4 Dynamo action
4.2.5 Conclusion
4.2.6 Appendix: Stabilization method
5 Momentum-based approximation of incompressible multiphase fluid flows 
5.1 Introduction
5.2 The model problem
5.2.1 The Navier-Stokes system
5.2.2 Level-set representation
5.3 Semi-discretization in time
5.3.1 Constant matrix diffusion on a model problem
5.3.2 Pressure splitting
5.4 Full discretization and stabilization
5.4.1 Space discretization
5.4.2 Stabilization by entropy viscosity
5.4.3 Compression technique for the level-set
5.4.4 Extension of the algorithm to the MHD setting
5.4.5 Finite elements/Fourier expansion
5.5 Analytical tests
5.5.1 Manufactured solution
5.5.2 Gravity waves
5.6 Newton’s bucket
5.6.1 Physical setting
5.6.2 Influence of Strain rate tensor
5.6.3 Influence of the surface tension
5.7 Free surface flow in an open cylinder
5.7.1 Physical setting
5.7.2 Numerics vs. experiment
5.8 Bubbles
5.8.1 Rising bubbles
5.8.2 Oscillating bubbles
5.9 Liquid metal droplet falling in a vertical magnetic field
5.9.1 Physical configuration
5.9.2 Falling droplet under gravity
5.9.3 Lorentz force as an external force
5.9.4 Full MHD setting
5.10 Conclusion
6 Conclusion and prospects 
6.1 Outcome
6.2 Outlook
7 Résumé en français 
7.1 Introduction
7.1.1 Contexte et motivations
7.1.2 Rappel des équations adimensionnées de la MHD
7.2 Le code SFEMaNS
7.2.1 Description du code
7.2.2 Développements récents
7.3 Viscosité entropique
7.3.1 Nécessité de modélisation
7.3.2 La viscosité entropique comme modèle LES
7.3.3 La viscosité entropique dans SFEMaNS
7.4 Application aux Simulations des Grandes Echelles (LES)
7.4.1 Application à des écoulements de Von Kármán
7.4.2 Application à des récipients cylindriques en précession
7.5 Approximation d’écoulements multiphasique avec la quantité de mouvement
7.5.1 Approximation numérique
7.5.2 Récapitulatif de quelques test numériques
7.6 Conclusion
7.6.1 Résultats
7.6.2 Perspectives
Bibliography 

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