Solving the Helmholtz equation – τ-method in radial direction

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Magnetohydrodynamics equations

When a conducting fluid is moving in the presence of a magnetic field, the contribution of σu × B in (1.4) must be taken into account. The coexistence of the electric currents and the magnetic field implies that the volume forces (1.12) must be considered in the Navier-Stokes equations (1.11). In such a case the Maxwell equations together with the Navier-Stokes equations could already be considered as the equations of magnetohydrodynamics. However, since magnetohydrodynamics mostly concerns the interaction between the fluid flow and the magnetic field then it is better to eliminate all other variables which are of less interest for a hydromagnetic problem. For identifying the influence of the magnetic field on the fluid, it is natural to choose (1.3) rather then (1.4) as the definition of j, because the fluid velocity u is not present in (1.3): j = 1 μ∇ × B.

Potential form of the MHD equations

Substituting (3.1) into (2.7) results in equations whose linear parts couple three fields (ψu,φu, p) and two fields (ψB,φB). Because of the semi-implicit numerical approach which we intend to use for time integration, the separation of variables in the linear parts of equations (2.7) is essential. It is easy to accomplish this decoupling given that the ˆez components curls of a vector field F = ∇ × (ψˆez) + ∇ × ∇ × (φˆez) have simple expressions: ˆez · F = −Dhφ.

Equivalence of potential and primitive variable formulation

Up to now we have not proved the equivalence between the potential and primitive variable formulations. Since we took the curl of equations (2.7a) and (2.7b) they gained an additional degree of freedomwhich wemust fix in such a way that these equations in potential form(3.10- 3.11) define the same velocity u and magnetic field B as the original MHD equations (2.7). We will first write (2.7) in a compact form, which will let us use a common form for (3.10) and (3.11).

Conductor/vacuum configuration

The case when the fluid of finite electric conductivity is restricted to a finite volume (here a cylinder) and is surrounded by vacuum is of special importance to us, because it models well the experimental configuration of the VKS experiment. As it was already explained in section 1.3.1, the boundary condition for the magnetic field can be simplified. We recall that the external magnetic field in vacuum satisfies ∇ × Bvac = 0 ⇒ B = ∇φvac.
In this case the continuity of all three components of the magnetic field are sufficient conditions to uniquely determine both the internal and external fields. The equations describing the internal field are given in (3.11) while the external problem is reduced to Dφvac = 0.

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Table of contents :

1 Introduction 
1.1 Motivation
1.2 Outline of this thesis
1.3 Magnetohydrodynamic equations
1.3.1 Maxwell equations
1.3.2 Navier-Stokes equations
1.3.3 Magnetohydrodynamics equations
1.4 Dynamo effect
1.4.1 Concluding remarks
I System description 
2 The von Kármán flow 
2.1 Brief overview of studies
2.1.1 History
2.1.2 Stability analysis
2.1.3 Turbulence
2.2 Mathematical model
2.2.1 System description
2.2.2 Dimensionless equations
2.2.3 Equations of magnetohydrodynamics
3 Poloidal-toroidal decomposition 
3.1 Motivation
3.2 Poloidal-toroidal decomposition and the gauge freedom
3.3 Potential form of the MHD equations
3.4 Compatibility condition
3.4.1 Equivalence of potential and primitive variable formulation
3.4.2 Hydrodynamic compatibility condition
3.4.3 Magnetic compatibility condition
3.5 Hydrodynamic boundary conditions
3.6 Magnetic boundary conditions
3.6.1 General case
3.6.2 Conductor/vacuum configuration
3.7 Discussion
3.7.1 Advantages/disadvantages
3.7.2 Concluding remarks
II Numerical method – Spectral solver 
4 Spectral discretization 
4.1 Introduction
4.1.1 Local methods
4.1.2 Spectral precision
4.1.3 Advantages and limitations of spectral methods
4.2 Azimuthal direction
4.3 Axial direction
4.4 Radial direction
4.4.1 Regularity condition
4.4.2 Regularization of an arbitrary spectral basis
4.4.3 Regular basis of radial polynomials
4.4.4 Differential operators
4.5 Discretization in 3D
4.6 Boundary condition regularization
4.6.1 Overview of singularity treatment techniques
4.6.2 Boundary velocity regularization
5 Spectral solver 
5.1 Introduction
5.2 Poisson solver
5.2.1 τ method – one dimension
5.2.2 Partial diagonalization method – two/three dimensions
5.2.3 Solving the Helmholtz equation – τ-method in radial direction
5.3 High order PDE solver
5.3.1 2D Navier-Stokes in streamfunction formulation
5.3.2 Multi-Poisson solver
5.4 Influence matrix
5.4.1 Green function method
5.4.2 Discrete Green functions method – influence matrix
5.4.3 Towards an invertible influence matrix
5.5 Tests – Stokes problem in 2D
5.5.1 Polynomial solutions
5.5.2 Non-polynomial case
5.6 Towards an MHD solver
5.6.1 External solution in vacuum
5.6.2 Continuity conditions
5.6.3 Multi-Poisson solver for induction equation
5.6.4 Influence matrix for the magnetic problem
5.6.5 Elimination of external solution
5.7 Concluding remarks
6 Stability/Validation 
6.1 Nonlinear term
6.1.1 Evaluation of −u × w
6.1.2 Regularity of the nonlinear term
6.2 Time integration
6.3 Tests
6.3.1 Axisymmetric rotor-stator configuration
6.3.2 First instability in 3D
6.4 Spectral convergence
6.5 Parallelization
6.6 Concluding remarks
7 Applications & perspectives 
7.1 Turbulent bifurcation
7.1.1 The effect of blades
7.1.2 Preliminary results
7.2 Axisymmetric turbulence
7.2.1 Theoretical framework
7.2.2 Experimental confirmations
7.2.3 Numerical results
7.2.4 Imposed axisymmetry
8 Concluding remarks 
A Recursive formulas for radial polynomials 
B Remarks concerningMPI parallelization 

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