Date : 31 Mars 2022 à 14h00, salle de réunion M2P2

Abstract : Nowadays a challenge remains the design of optimized plasma scenarios for tokamak operation to control the heat flow from the core region to the wall. This calls for the development of efficient and reliable numerical codes with predictive capabilities for plasma simulations. Despite significant progress the last years, predictive capabilities of current transport codes are still acknowledged by the international community as being largely insufficient. Among the new capabilities for the codes expected to get predictive capabilities, the accurate discretization of real geometries of tokamak chambers, the flexibility with respect to the magnetic geometry and the computational efficiency in terms of speed have been yet clearly identified by the fusion community. To solve all these issues, a new approach is proposed in the present work which aims to develop a high-order finite elements code based on hybrid discontinuous Galerkin numerical scheme and an efficient implicit time integration method for solving non isothermal Braginskii reduced fluid equations in versatile tokamak and magnetic equilibrium geometries. The use of such numerical scheme allows to perform simulations with time evolving magnetic configurations, avoiding expensive re-meshing of the computational domain. The first part of the thesis shows the structure and the realization of such a numerical tool. The feasibility of the latter is then investigated through a careful validation and benchmarking operation with SolEdge3X. The physical model is then enriched with self-consistent sources of particle due to plasma recycling and sources of energy due to Ohmic heating. In particular, a fluid model for neutrals is implemented and coupled to the plasma by ionization-recombination-radiation terms. The introduction of the sources allows to perform 2D simulation of a domain of computation which encompass a full poloidal cross section of the tokamk. The capability of such a model of reproducing the main features of a detached plasma is investigated in the second part of this work for the WEST tokamk machine. Then the first core-edge transport simulations of an entire WEST discharge (shot #54487) are shown from the start-up phase to the final plasma landing. The experimental magnetic equilibrium which evolves from a high field side limiter to a X-point configuration is implemented in the simulation together with the experimental gas puff time evolution. Comparisons between experimental interferometry and synthetic simulation data show a remarkable agreement in the plasma center with an accurate prediction of the ramp up and the flat top phase of the central lines integrated density. The agreement is less good however in the vicinity of the X-point. Finally, the time evolution during the discharge of the particles and heat fluxes at the wall, are analyzed showing in particular the energy redistribution between ion and electron during the discharge. The present result are also used to assess the tungsten sputtering, using both a simple cinematic model and the impurity tracker monte-carlo code ERO2.0. The analysis confirms the need to consider full discharge simulation to accurately treat the W source of contamination. The work also demonstrates the interest of developing magnetic equilibrium free solver including efficient time integration to tend toward predictive capabilities in the future fusion operation. 

Doctorant : Ahmed HODAIB 

Date de soutenance : le 15 mars 2022 à 15h00 ; Amphi 3 Centrale Marseille

Abstract  : In aircraft engines, a secondary air flow is obtained from an intermediate compressor stage, to be used to cool the turbine disks. This flow passes through the high-pressure compressor inter-disk cavities (Farthing et al., ASME J. Turbomach., 1992). A better understanding of this complex buoyancy-induced flow is essential to determine the thermal stresses, the radial growth of the blades, due to thermal expansion, and the temperature rise of the air used for cooling. Besides, to be able to determine the optimum clearance between the rotating blades and the surrounding casing, in order to improve the engine performance. This convective flow is not only unsteady and three-dimensional, it is unstable. Due to high temperature differences, the flow and heat transfer give rise to a strongly conjugate problem: the flow is affected by the temperature of the disks, and vice versa (Owen & Long, ASME J. Turbomach., 2015). The compressible Navier-Stokes equations, coupled with the energy equation and perfect gas law, are solved in the framework of the Low Mach Number (LMN) approximation, allowing a reduction of computational costs by filtering the high-speed waves while keeping a good accuracy by considering the compressibility effects (Motheau & Abraham, J. Comput. Phys., 2016). A fourth-order compact spatial discretisation scheme combined with parallelised Fourier method is implemented on a staggered grid. A second-order semi-implicit scheme is introduced for time integration. A two-step algorithm is developed for the solution of the LMN equations. In a first step, the thermodynamic variables are calculated through an iterative process, and used to compute the velocity divergence. In a second step, the variable density continuity and momentum equations are solved using a projection-type method. A parallelized iterative domain decomposition technique is implemented for the simulation of the three-dimensional flow and heat transfer in a T-shape model cavity. The parallelisation of the resulting computational code is performed through a hybrid MPI/OpenMP approach. Spatial and temporal accuracies of the proposed algorithm are checked on a manufactured solution in a simplified configuration. Then, the algorithm is applied to study the flow and heat transfer in an idealised compressor inter-disk cavity, while considering conduction inside the walls, to allow for a more accurate thermal balance. The results are compared with data available in the literature based on local Nusselt numbers. A parametric study is done for a range of the two main parameters governing the flow, according to Farthing et al. (ASME J. Turbomach., vol. 114, pp. 229-236 and pp. 237-246, 1992): the temperature difference and the Rossby number. To include surface radiation exchanges, the discrete radiative heat transfer equation is solved based on the zonal method. The adequacy of the proposed Low Mach number approach is shown, compared to Boussinesq approximation. Moreover, the validity of neglecting the gravitational acceleration with respect to the centrifugal acceleration in the equations is discussed. Then, the definition of an effective Rayleigh number is established, where both centrifugal and gravitational accelerations are taken into account in the buoyancy terms. The results reveal that the flow exhibits a Poiseuille-Rayleigh-Bénard-like instability, and that this effective Rayleigh number governs the flow structure and the heat transfer in the whole cavity, and hence the stability of the flow. In the end, it is shown that radiative exchanges become more significant the more we get closer to the inner radius of the cavity, in agreement with the results reported by Tang & Owen (ASME J. Turbomach., 2021). It is observed that the temperature profiles at the upstream and downstream disks approach each other, when radiation is considered, where the upstream disk temperatures increase. 

Doctorant : Fabriel FARAG

Date de soutenance : le 4 février 2022 à 14h00 ; Amphi 3 Centrale Marseille

Abstract  : Since the late 1970's, computational fluid dynamics solvers became essentials due to increasingly complex applications requiring fluid solutions. The small scales necessary for industrial applications often need a very fine grid or very small timestep. This dramatically increases the computational cost of nowadays simulations. To design more computationally efficient solvers, a popular approach is to use Lattice-Boltzmann methods. Originating from the kinetic theory of gases, this method have gained a tremendous popularity among fluid dynamicists due to its cheap and easily implemented collide & stream algorithm. However, its intrinsic assumptions confines classical Lattice-Boltzmann solvers to weakly compressible flows. Yet, some compressible models have been proposed. The purpose of this manuscript is to improve the robustness as well as accuracy of compressible Lattice-Boltzmann models. To this end, the Lattice-Boltzmann method is fully reinterpreted as a numerical scheme. This allows a straightforward and parsimonious derivation of the equivalent Navier-Stokes-Fourier system using the sole assumption of a negligible timestep. Using this formalism, the order of accuracy is shown to depend on the collision kernel, as well as the mechanical constitutive model. Various models are investigated and we show that the Knudsen number is not the sole parameter controlling the consistency with the Navier-Stokes-Fourier model. Additionally, capabilities of the entropy equation to model low supersonic flows is explained through standard shock wave theory arguments. A MUSCL-Hancock scheme is employed to discretize the entropy equation and improve both stability and accuracy compared to previous schemes. Equipped with this new formalism, a compressible pressure-based model is proposed and validated on various supersonic test cases. Then, we unify all compressible models proposed by our group under a single formalism and investigate the differences and optimal choices for the various degrees of freedom of our family of models. Finally, this unified model is validated on high supersonic smooth flows and low supersonic shocked flows.