1.2 Energy and particle confinement/transport

1.2.1. Turbulence and transport phenomena

Background and Objectives: In this activity we study drift-type turbulence, large-scale flows and transport phenomena in tokamaks. Although there have been many advancements in the basic understanding of the physics that govern ion and electron heat, momentum and particle transport, there are still many unexplained phenomena that still need to be studied in detail. The analysis of novel experimental results of high performance tokamak plasma discharges has lead to the conclusion that plasma turbulence driven from spatial gradients is the dominant mechanism for the anomalous transport in tokamaks. Furthermore, macroscopic flows, coherent structures or large-scale events (generated by plasma turbulence) may lead plasma to a state of intermittency regulating the transport. Therefore, the description of the dynamics of plasma instabilities and the associated turbulence becomes crucial for the understanding of transport in tokamaks. The main goal is to develop, extend and investigate (both analytically and numerically) advanced models of plasma turbulence, which will be able to describe the complex dynamics of plasma turbulence and justify the features of the observed transport in tokamaks. One of the fundamental tools that can be used to describe the turbulent fluctuations is the determination of high order statistical moments. In order to investigate complex dynamics related to anomalous transport under the presence of turbulent fluctuations as well as coherent structures and large-scale flows, a variety of Hamiltonian models can be used for analytical and numerical studies, taking advantage of powerful techniques. In addition, the description of the anomalous transport in tokamak plasmas using the gyrokinetic formalism has been of great interest, in recent years. Gyrokinetic calculations and their comparison with the experimental database can be used in order to extend the understanding of transport phenomena and to further develop the theoretical framework of the gyrokinetic formalism.

Work performed in year 2008 (in co-operation with the Institutes indicated):

1. We analysed results obtained by numerical simulations of the Hasegawa-Wakatani drift wave turbulence using a two-dimensional fluid code, and of the electromagnetic resistive ballooning turbulence using the EMEDGE3D global code. We investigated the properties of kurtosis and skewness associated with plasma edge fluctuations and found similar results to the scaling relation observed in the fluctuation signals measured in TORPEX. Furthermore, by constructing a simple non-Gaussian process we derived the same scaling relation within the 95% confidence bounds of the least-square fitting. More details can be found in Annex 1. (In co-operation with CEA.)
2. Stability analysis of TITG mode using the GYRO code including parallel dynamics: This task has been withdrawn because of reduced availability in 2008 of the scientist in charge.
3. The techniques for constructing Hamiltonian mappings of either implicit or explicit form have been further developed in order to include perturbations of a more general analytical form. In addition, these techniques have been extended in order to be applicable in cases where the perturbations are not known analytically, but given as a list of measured values, as well as in cases where we have stochastic perturbations. (In co-operation with ULB, MEdC, TEKES.)
4. Stochastic particle motion has been considered in terms of a perturbed guiding-centre Hamiltonian. The unperturbed part of the Hamiltonian describes particle motion in an axisymmetric equilibrium, in a canonical variable set, related to magnetic coordinates. A list of cases for perturbations of interest corresponding to stochastic static and/or wave fields has been formulated. A preliminary comparative study of different methods for investigating the interplay between the intrinsic complexity of the particle motion with the external stochasticity due to static and/or wave fields has been performed. (In co-operation with ULB, MEdC, TEKES.)
5. A fully nonlinear, energy conserving, six moment drift fluid model has been developed. The drift fluid equations are directly derived from a drift fluid Lagrangian with constraints on the density and temperatures. The Lagrangian nature of the problem leads to the fact that the final equations conserve energy exactly and are fully nonlinear. The model is a first step to deriving a fully nonlinear, energy conserving gyrofluid model that can be used in numerical computations.
6. The gyrokinetic Vlasov code GKW has been further developed to include Coriolis drift effects also in the direction parallel to the magnetic field. Benchmarks have been made both for the linear regime with the gyrokinetic codes GENE, GS2 and GYRO, and for the nonlinear regime with the gyrokinetic code GS2 for the cyclone base case as well as the Waltz standard case. The code has also been extensively tested in comparisons with dedicated experiments on momentum transport at JET and also benchmarked with the Weiland model. An interface between GKW and the JET database has been developed, including direct input of EFIT parameters. More details can be found in Annex 2. (In co-operation with University of Warwick and EFDA JET.)
7. Various numerical simulations have been made for the comparison of gyrokinetic code results with the JET experimental results. The gyrokinetic codes GS2, GKW, GYRO and GENE have been used. From the results so far it is apparent that the gyrokinetic codes predict very well the threshold of the experimental ITG mode. To study the ITG stiffness many nonlinear runs have been performed which show that the stiffness, at least in low rotation cases, agrees well with the experimental results. However the nonlinear threshold does not agree with the experimental one due to the existence of the Dimits shift. (In co-operation with EFDA JET.)

1.2.2 Particle transport model

Background and Objectives: In order to simulate realistic problems within the tokamak, one needs to know the charged particle transport properties of the plasma fluid model, which in this case, is of the deuterium and tritium. To achieve this, one option is to calculate beforehand the particle transport properties from the literature based on experimental and numerical results, and to incorporate them in the existing plasma fluid model. Another option is to solve the charged particle conservation equations of momentum and energy, that aid in the much more accurate calculation of the transport properties of the plasma fluid model. Furthermore, to be able to analyse neutral gas heating effects, it is necessary to incorporate the mass, momentum and energy conservation equations into the plasma fluid model, and couple these equations with the already existing charged particle continuity and Poisson solvers. The objective of this work is to lessen the dependency on transport properties from other numerical and experimental results by solving additionally the conservation of momentum and energy equations for the charged particles. This will provide the ability to simulate collisional plasmas in the tokamak of ITER by solving the classical Brangiskii fluid equations, thus achieving a much more accurate estimate of transport properties parameters. This goal is expected to be achieved within 2009 and 2010.

Work performed in year 2008:

1. The Brangiskii equation was identified as the most appropriate and has been implemented, as described in the following.
2. The compressible Navier-Stokes solver developed in 2007 comprises of the mass, momentum and energy conservation equations that describe the neutral gas particle dynamics. Furthermore, the Poisson and charged particle continuity equations solver describes the charged particle dynamics within the plasma. Consequently, the first step completed in 2008 was the coupling of these two models and the study of the interaction between the charged and neutral gas particles. The coupling of these two models, which is based on the incorporation of mass, momentum and energy conservation in the plasma fluid model, was implemented through source terms, such as Joule heating, exchange of momentum and energy due to elastic and inelastic collisions, and generation and production loss processes, as well as through the ratio of the electric field to the neutral gas density per unit volume.
3. To validate the coupling of the two solvers, the avalanche and streamer propagations in a 1 mm DC parallel plate gaps, as well as in RF 40 MHz in 1 cm parallel plate gaps have been successfully simulated using the developed adaptive mesh generator. This has demonstrated the successful coupling of the Navier-Stokes solver with the charged particle continuity and Poisson solver. Therefore, no more work was conducted, since the mass, momentum and energy conservation equations have been successfully incorporated into the plasma fluid model. Further details are presented in Annex 3.

1.3 MHD stability and plasma control

1.3.1. ECRH for MHD control

Background and Objectives: This activity focuses on the improvement of the theoretical understanding and the interpretation of experiments on MHD instability control with the use of EC heating and current drive. The activity consists of two parts:

In the first part, the main target is the derivation of a self-consistent numerical model for NTM stabilisation and control with ECRH/ECCD. The primary attempt includes the coupling of different kinds of wave propagation solvers (ray tracing, beam tracing and full-wave) with the modified Rutherford equation or more sophisticated MHD models for the description of the NTM dynamics in realistic magnetic geometry, including the non-axisymmetric perturbations (in the form of magnetic islands) generated by the NTMs.

For the second part, an experimental activity in ASDEX Upgrade has been planned, in order to assess the theoretical model regarding the advantage of early ECCD for the suppression of NTMs. More specifically, the experimental study was aiming at the observation of the three operational regimes of early ECCD (predicted by the theoretical model for “sufficiently” broad deposition profiles), which involve: (i) the complete suppression of the NTM (at maximum ECH power); (ii) the partial suppression of the NTM with the advantage of the early application (at reduced ECH power), and (iii) the partial suppression of the NTM without the advantage of the early application (at low ECH power).

Work performed in year 2008 (in co-operation with IPP-Garching):

• (iv) For the prescription of a non-axisymmetric geometry corresponding to magnetic islands, a first approach was made in terms of the tokamap. After a detailed investigation, the tokamap was proved ineffective in providing a consistent description of the tokamak magnetic field [see the detailed reports in tasks 1.5.1(ii) and 3.4.2(v)]. A different model was then used, known as “magnetic field-line tracing”, where the field lines are calculated as Hamiltonian trajectories, the toroidal flux and poloidal angle being the canonical variables, the poloidal flux being the Hamiltonian and the toroidal angle playing the role of time. Before using this geometry as a background for the coupling of our ray tracing code with the modified Rutherford equation, and with main purpose to understand deeper the electron dynamics in the perturbed magnetic geometry (which is directly connected to the ECCD efficiency), we initiated the numerical study of electron orbits in the magnetic field emerging from magnetic field-line tracing and in the presence of a localised EC beam.
• (v) The experimental programme, which was endorsed for 2008 by the ASDEX Upgrade programme committee, included comparisons of early (i.e. before the growth of the NTM) and late (i.e. after the saturation of the NTM) application of ECCD, in identical high-beta plasmas. Substantial experimental time (more than originally requested) was dedicated to the following activities: (a) The calibration of the new ECRH system, involving a multi frequency gyrotron with steerable mirrors, which was used for the first time in the NTM programme. (b) The establishment of the so-called “NTM reliable shot-scenario”, i.e. the determination of the discharge conditions at which stationary (in frequency and amplitude) NTMs could be obtained. In that respect we deployed every available technique for creating alternative plasma conditions, such as boronisation of the first wall for reducing the light impurity content, nitrogen seeding for radiation cooling of the divertor, standard H-mode discharges at 0.8 MA and improved H-modes discharges at 1 MA, etc. The conclusion was always that the NTMs, which could be routinely destabilised by 12.5 MW of NBI, could not reach the sustainment phase (which was required for the application of ECCD), but were degrading despite the otherwise stationary plasma conditions. It was finally agreed that the experimental project will be continued (as originally planned) only after this unexpected NTM behaviour is understood, or when the new EZ4 power generator (which will allow 20 MW of NBI) is commissioned and the defected tungsten tiles are replaced at the winter opening. It is also understood that this unexpected behaviour of the NTM must be somehow linked to the tungsten wall, which last year replaced the carbon wall, but the matter is still under investigation.

1.3.2. Development of a two-fluid MHD solver for a toroidal geometry with a general cross-section

Background and Objectives: In magnetic fusion research, the magnetohydrodynamic equations have been used quite successfully to study the equilibrium and stability of flows in fusion plasmas. Nonlinear MHD simulations are powerful for the study of the evolution of ideal or resistive instabilities located inside or at the edge of the fusion plasma. This type of simulations is able to capture various fast growing MHD instabilities that may result in severe destructions of tokamak confinement. Only a few codes worldwide can handle together the non-linear, compressible, two-fluid and thermal MHD equations for fusion plasmas, confined in generalized cross-section domains. The main objective of this task is to provide the Fusion Community with such a European code.

Work performed in year 2008 (in co-operation with CEA and ULB):

(i) The cylindrical and incompressible version of the MHD code named SFELES was finished and a flow driven by an external electromagnetic force was successfully tested. An analytic solution of this flow was also derived. We focussed on the development and testing of the fully compressible version of the code SFELES for the toroidal geometry with an arbitrary-shaped profile. For this purpose the well-known tearing instabilities have been used as a test case (see Annex 4).

1.3.3 Effects of rotation on stability of multi-phase MHD turbulence

Background and Objectives: The liquid metal circulating through the reactor blanket is exposed to strong magnetic fields. While magnetic fields tend to suppress turbulence, curvature and system rotation and fringing magnetic fields can, under certain conditions, destabilise the flow and lead to turbulence augmentation. It is therefore important to understand the combined effects of mean shear, strong magnetic fields and system rotation on transport rates in the liquid metal flow. The goal of this activity is to examine the transport of a dispersed phase (particles) in the presence of strong magnetic fields, mean shear, and system rotation using a fully-3D spectral code. (This is a multi-annual activity performed in cooperation with ULB).

Work performed in year 2008:

(i) Lagrangian particle transport was implemented in a fully-3D MHD spectral code. The necessary code modifications were implemented for the simple case of an initially isotropic field exposed to a strong magnetic field at low magnetic Reynolds number. A series of simulations was carried out to investigate preferential concentration of the dispersed phase depending on the combination of key dimensionless parameters (see Annex 5). In a recently published paper [D. W. I. Rouson, S. C. Kassinos, I. Moulitsas, I. E. Sarris, and X. Xu, Physics of Fluids 20, 025101 (2008)], the “Dispersed Phase Structure Dimensionality” (DPSD) concept was introduced as a new one-point statistical measure of preferential concentration. In a series of direct numerical simulations of particle dispersion in MHD turbulent flow, the DPSD was shown to provide an accurate description of the spatial distribution of the dispersed phase. This highlights the potential of the DPSD concept to be used as a flow diagnostic and for model development.

The simulations completed in 2008 show that the degree of dispersed-phase structural anisotropy depends on the particles’ response times and the time interval over which the magnetic field is applied. Particles with short response times exhibit no significant structural anisotropy. Particles with very long response times exhibit large but transient structural anisotropy. In between these extremes, structural anisotropy is characterised by the accumulation of particles in parallel sheet-like structures.

Given the successful implementation of Lagrangian particle transport in the non-deforming grid case (initially isotropic condition), and the demonstration of the potential of the DPSD concept, further modifications of the full-3D spectral code were initiated. These modifications will be completed in 2009 and will allow the study of Lagrangian particle transport in MHD flows in the presence of mean shear and system rotation.

1.5 Physics of plasma heating and current drive

1.5.1. ECRH-ECCD and transport in fusion plasmas

Background and Objectives: In this activity we investigate the EC wave propagation, absorption and current drive, as well as the wave-electron resonant interaction, in plasma geometry relevant to ECRH/ECCD experiments for heating, diagnostics and NTM control. The activity has several parts:

In the first part [task (i)], we focus on the importance of nonlinear wave-particle interaction, hot plasma response and wave/beam effects in the propagation and absorption of EC waves in tokamak plasmas. In the field of wave propagation in plasmas, the mainstream in theory and applications is oriented to frequency-domain asymptotic methods assuming linear plasma response, where the solution of the wave-plasma system presents less difficulty. However, in many cases of interest (like e.g. O-X-B mode conversion, high-power ECRH) this approach breaks down, the solution becomes questionable and therefore a more sophisticated treatment (full-wave and nonlinear) is called for. Our target is to upgrade the physical schemes used to study the wave propagation, absorption and current drive in fusion plasmas.

In the second part [task (ii)], we analyse the propagation of EC waves, their absorption, the current drive and the transport of particles when the NTM instability is excited inside a tokamak. The general idea for the stabilisation of NTMs is to drive current (ECCD) in the vicinity of the magnetic island’s O-point. So far, the analysis of the EC propagation and absorption was done in axisymmetric magnetic topology, ignoring the presence of islands. We study the propagation in the presence of magnetic islands, focusing on the effect of the island topology on the efficiency of the EC absorption and ECCD. The plasma response is either assumed linear (plasma dielectric tensor) or calculated by statistical methods (ensembles of particle motions in the perturbed magnetic field).

In addition, in the third part [tasks (iii)-(vii)] nonlinear dynamics of wave-particle interactions in fusion plasmas are considered from the point of view of applications to Electron Cyclotron Resonant Heating (ECRH) and its role in the stabilization of the Neoclassical Tearing Modes (NTM), Electron Cyclotron Current Drive (ECCD) and plasma diagnostics through interactions with rf pulses. The Hamiltonian formalism along with a set of accompanying tools such as the phase space analysis, the symplectic integration, the Canonical Perturbation Method and Lie transforms are used in order to extend the theory beyond the quasilinear approximation for realistic fusion plasma configurations and general form of the wave spectra.

Finally, in the last part [task (viii)], the main objective is to address the advantages of the utilisation of non-Gaussian beams in ECRH systems. This will be performed in terms of the modelling of ECRH/ECCD in simplified and realistic tokamak geometry, using asymptotic methods (ray tracing, paraxial WKB beam tracing) for following the wave propagation and the linear hot plasma dielectric tensor for determining the plasma response. In collaboration with IPP-Garching, we study the evolution of non-Gaussian beams in terms of the paraxial WKB beam tracing method. The results are exploited in order to upgrade certain features of the existing TORBEAM code, such as: a) The propagation and absorption of arbitrary-shaped EC beams in realistic magnetic configuration, b) The description of the beam evolution, in terms of the generation of higher-order Gaussian-Hermite modes, in scenarios where the effect of localised and/or asymmetric absorption is significant.

Work performed in year 2008 (in co-operation with the Institutes indicated):

(i) We continued the development of a full-wave code for the description of the propagation and absorption of EC beams in tokamak geometry (see Annex 6). The code under development consists of three basic parts: (1) The wave propagation solver, based on the FDTD algorithm, (2) the computation of the magnetic equilibrium, based on a Hamiltonian model for magnetic field-line tracing, (3) the calculation of the plasma response to EC waves, based on different linear/quasilinear/nonlinear models. We have completed and benchmarked the first and the second part, whereas for the third part, at this point, we can use only the physics models provided by linear EC theory (up to weakly-relativistic hot plasma dielectric tensor). Regarding the development of the routine for the calculation of the plasma response, a feasibility study is being performed on the applicability of the available quasilinear and nonlinear models for the case of interest.

(ii) We initiated the analysis of the propagation, absorption and current drive of EC waves in tokamak plasma in the presence of NTMs. After a thorough investigation, the tokamap has been proved ineffective in providing a concrete model for the calculation of the tokamak magnetic field. The tokamap allows calculating the magnetic field-lines, but not the magnetic field itself [see the more detailed report in task 3.4.2 (v)]. A new model was then used, where the magnetic field lines are calculated as Hamiltonian trajectories, the toroidal flux and poloidal angle being the canonical variables, the poloidal flux playing the role of the Hamiltonian and the toroidal angle the role of time. We have completed the ray-tracing code that calculates the propagation in the magnetic geometry provided by the aforementioned Hamiltonian model, for plasma response according to the linear EC theory, i.e. cold plasma dielectric tensor for the propagation and hot plasma absorption coefficient for the damping. For more details see Annex 7. (In co-operation with IPP-Garching.)

(iii) The work on wave-particle interactions taking into account the inhomogeneity in the confining magnetic field (initiated in the previous period) has been continued and a novel approach on the quasilinear theory has been developed. As a result an action-diffusion equation has been obtained for wave-particle interactions in an axisymmetric toroidal magnetic field. This diffusion equation describes particle momentum and radial transport due to interaction with rf waves. The corresponding diffusion tensor is a non-singular tensor describing both resonant and non-resonant interactions (Annex 8). (In co-operation with PSFC-MIT, USA.)

(iv) The quasilinear theory has been extended to cases where magnetic islands (due to NTMs) are present. The corresponding diffusion tensor contains information related to the combined effects of both the rf-wave packets and the axisymmetry-breaking magnetic islands on particle momentum and radial transport (Annex 8). (In co-operation with PSFC-MIT, USA.)

(v) The quasilinear theory has been extended in the relativistic regime. The relativistic effects have been shown to modify the unperturbed Hamiltonian of the particle motion in the axisymmetric magnetic field and the corresponding resonances between the unperturbed particle motion and the rf-wave and/or magnetic islands perturbations. More details can be found in Annex 8. (In co-operation with PSFC-MIT, USA.)

(vi) The quasilinear models developed previously have been further considered for coupling to ray-tracing code and preliminary feasibility studies have been performed. (In co-operation with PSFC-MIT, USA.)

(vii) The study of wave-particle interaction for localised wave packets which are obliquely propagating with respect to the confining magnetic field has been initiated. Preliminary results show a crucial dependence of the interaction on the form and width of the wave packets and a significant role of finite Larmor radius effects. (In co-operation with PSFC-MIT, USA.)

(viii) In this period we have in addition completed the coupling of our FORTRAN routine NGBT, which computes the propagation and absorption of non-Gaussian EC beams based on a model where the beam amplitude profile is described as superposition of Gaussian-Hermite modes, with the TORBEAM code, which computes the propagation, linear absorption and current drive of Gaussian EC beams in tokamak geometry for arbitrary launching conditions and analytic or experimental magnetic equilibrium (see Annex 9). The result gives TORBEAM the ability to simulate the propagation of arbitrary EC beams in tokamak geometry.

1.5.2 Study of heating effects

Background and Objectives: During 2007 and early 2008, the incorporation of the plasma heating effects in the plasma multiphysics code was successfully completed. Before the multiphyscis code can be used one needs to validate that the neutral gas effects, such as pressure and temperature, are working properly within the plasma multiphysics software. For this reason, it is necessary at a first stage to simulate a discharge where neutral gas heating effects are included, but do not affect the development of the discharge, and verify that this is indeed the case. At a second stage, it is necessary to test whether the neutral gas heating effects, when coupled with the charged particle continuity and Poisson solvers behave properly, and this can be achieved by simulating a realistic gas discharge problem where there is real coupling between the charged and neutral gas particles.

Work performed in year 2008:

(i) The ability of the code to analyse discharges in different geometries such as short and long gaps under different applied voltages, and geometric configurations, in two-dimensional axisymmetric coordinates with heating effects included has been demonstrated. This has been achieved by simulating successfully the avalanche and streamer propagation in long point-plane gaps of 5 cm apart with the point electrode being a hyperboloid of radius 50 µm. In the above discharge, the avalanche and streamer formation and propagation have been simulated and branching streamer phenomena have been observed in agreement with the literature. Parameters such as streamer velocities, charged particle densities, avalanche and streamer development characteristics, formation times and distances, radial and axial electric fields, have been simulated during the development of these discharges.

(ii) At a second stage, in order to analyse and study the effects of pressure and temperature of the gas and the coupling of the newly developed software in real heating applications, the plasma neutral gas heating effects have been successfully simulated during the post-streamer discharge in short DC parallel plates. In the above discharge, it has been demonstrated that the numerical model developed is capable of analysing the neutral gas heating effects in intense activity regions, such as those of the cathode fall and negative glow regions within the normal and abnormal glow discharge, where extremely intense electric fields exist. Further details are presented in Annex 10.

1.6 Energetic particle physics

1.6.1. Alfvén wave-particle interactions at sub-Alfvénic velocities

Background and Objectives: The aim of this activity is to develop a coherent theory capable of describing the experimental results on sub Alfvénic resonances observed in JET. In particular, the interest lies in the understanding of the behaviour of α-particles originating from nuclear reactions in deuteron–deuteron or deuteron–triton thermonuclear reactions, or due to injection of neutral beams.

Work performed in year 2008 (Partly in co-operation with JET):

(i) The expression for the equation of equilibrium has been derived and numerical integrations are needed to visualise its content. New results have been obtained from the quantisation of the current perturbation. In particular, it has been shown how an elliptically polarised Alfvén wave can simultaneously increase both the current density in a tokamak and the power transferred to the alpha-particles-ions in general. At the same time the transfer rates were given for thermal (Maxwellian) plasmas as well as for non-thermal ones, since it seems very probable that certain areas play host to non-thermalised distributions. (This activity has been terminated on 30 April 2008.)

1.7 Theory and modelling for ITER

1.7.1 Development of computational fluid dynamics solvers for liquid-metal flows relevant to blanket modules (DEMO incl.)

Background and Objectives: The reactor blanket and the circulating liquid metal contained within are exposed to the strong magnetic fields used to confine the plasma. When the liquid metal enters or exits the area of influence of the magnetic field, the forces acting upon the liquid metal change, which in turn can modify the characteristics of the flow. The goal of this activity is to model the flow of liquid metals in strong magnetic fields, and examine the effects of fringing magnetic fields and of the electrical conductivity of the blanket material on the flow. The modelling is done using a parallel, unstructured, Navier-Stokes solver developed in collaboration with ULB.

Work performed in year 2008 (in co-operation with ULB):

(i) A FORTRAN 90/95 module for modelling MHD flows was developed in collaboration with ULB in 2007. The solver was validated for hydrodynamic and low Hartmann number flows in 2007. In 2008, the solver was validated for the different high Hartmann number flows as described below. Further details are given in Annex 11.

(ii) The code was verified for flow in a pipe at bulk Reynolds number of approximately 8000 and Hartmann numbers up to 7000 for both perfectly conducting walls and perfectly insulating walls. Simulation results indicate that the conducting pipe has a grater shear stress than the insulating pipe at the same Hartmann number due to the braking action of the Lorentz forces induced by the wall electric currents. The solver was also verified for flow in a pipe with perfectly insulating walls at Reynolds number of 8000 and a fringing magnetic field with maximum Hartmann number of 7000 that falls to zero along the stream-wise direction. Simulation results for the insulating case in the fringing magnetic field indicate that turbulence can be triggered by the shearing action of the jets causes by the Lorentz forces.

(iii) Two available codes for the simulation of three-dimensional MHD duct flows, a mixed finite elements/spectral code and a finite-difference code, were tested for their capability of performing under high Hartmann numbers. The major difficulties were encountered in the narrow regions of the Hartmann and the side layers. In the case of high Hartmann numbers, turbulence may appear and the resolution may be very poor. We are working on the problem of the increased resolution, which is important for the turbulent duct flow simulation under high Hartmann numbers, by modifying the CFD codes to operate in the newly developed PC cluster at the University of Thessaly (see Annex 12).

1.7.2 Development of computational fluid dynamics solvers for viscous MHD flows

Background and Objectives: The study of conductive and viscous fluid flow and transport phenomena in magnetic fields is of fundamental interest for the understanding of the underlying physics. Due to many particular problems related to the magnetohydrodynamic flows, for example, MHD turbulence, heat transfer, transition and stability, a series of individual accurate solvers based on computational fluid dynamics techniques are being developed.

Work performed in 2008 (in co-operation with the Institutes indicated):

(i) Continuing the work of previous years, non-uniform grids were implemented into the CFD models and numerical acceleration schemes for high Hartmann MHD duct flows were tested. Non-uniform grids were used in the CFD codes and efforts were made to implement numerical acceleration schemes for high Hartmann MHD duct flows. For more details see Annex 12. (In co-operation with ULB.)

(ii) The effect of the magnetic field strength and direction in turbulent channel flows has been determined using accurate DNS simulations. The effect of the Prandtl number on the heat transfer was assessed. The free convection flow and heat transfer in cylindrical domains was studied in detail. The effect of the magnetic field on the transition from an initial 2D flow to a 3D flow was also assessed. The study of free convection MHD flow in closed cylindrical domains has been continued for the case of non-uniform temperature profiles. For this case, the initial axisymmetric flow field becomes three-dimensional under the effect of transverse magnetic field. The turbulent statistics of the MHD channel flow were calculated and the effect of the direction of the magnetic field on them was assessed. See Annex 13 and Annex 14 for more details. (In co-operation with ULB.)

(iii) Calculations of critical wavenumber and Grashof number have been performed for different Ha numbers for free convection flow in cavities with orthogonal cross section. It was thus shown that three dimensional disturbances are more unstable that 2D ones, leading to a lower critical Gr number. Benchmark calculations and the relevant eigenvectors are presented in Annex 15. In addition, the finite element formulation for Taylor-Couette flow has been applied in cylindrical geometry, in the presence of an axial magnetic field and preliminary computations have been performed (see Annex 16). Calculations were also performed on the base solution of forced flow in a duct in the presence of a strong magnetic field, Ha of the order of 1000. Work progress in the stability of this type of flow is mainly bibliographical in order to assess the open issues regarding transition and the effect of side and Hartmann layers. (In co-operation with ULB and FZK.)

(iv) This task was completed in its present state. A PhD was awarded to Dr D. Fidaros (2007) and a related article was accepted for publication.

1.7.3. Development of an immersed boundary solver for MHD flow for blanket modules (DEMO incl.)

Background and Objectives This activity aims in developing an Immersed Boundary (IB) methodology for MHD flows of liquid metals in complex geometries. Such a technique can extend the predictive capability of existing flow solvers and the range of computable wall-bounded MHD flows at a reasonable cost.

Work performed in year 2008 (in co-operation with ULB and FZK):

(i) The IB method has been successfully extended to account for immersed non-conducting surfaces and was implemented in a previous hydrodynamic IB code. A projection scheme for current density has been developed which is appropriate for MHD simulations in complicated geometries. Extensive tests have been performed for laminar two-dimensional MHD flows in square ducts with non-conducting walls at high Ha numbers, in order to verify the methodology and the numerical implementation. Investigation of the flow around circular cylinders has been initiated as a more challenging benchmarking case. Details are presented in Annex 17. The new IB MHD simulation code was further tested and validated to ensure numerical stability, accuracy and algorithmic correctness. For this purpose, the laminar MHD flow around circular or rectangular cylinders was considered as a benchmark problem in order to verify the method for various intensities and directions of the magnetic field. In order to further explore the developed extension of the method to complicated three-dimensional problems, a series of fully developed turbulent MHD simulations in plane channels or ducts with or without solid obstructions were conducted. These challenging problems have not been widely studied, due to the increased computational cost in traditional computational approaches. The constructed extension of the IB method proved that it is a valuable numerical tool for the close examination of these cases at a reasonable cost. Finally, preliminary calculations for the MHD flow around a circular cylinder confined in a rectangular duct, indicate the appearance of very interesting physical phenomena where little is known.