A Godunov-type method in Lagrangian coordinates for computing linearly-perturbed spherically-symmetric flows of gas dynamics

An explicit Godunov-type method in Lagrangian coordinates is devised for computing three-dimensional linear perturbations about spherical radial flows of gas dynamics. This method relying on a description of the perturbed flow in terms of linear Lagrangian perturbations is an outgrowth of an unpublished work by the author (Clarisse, 2001) and of the Godunov-type method for multi-material flows in planar symmetry presented in (Clarisse et al., 2004). The principle of a discrete formulation of the geometric conservation law (Thomas 1979) for the motion perturbation is introduced, granting mass conservation at the perturbation level. A practical time-step constraint for the numerical stability of the linear perturbation computation is provided in the case of third-order non-degenerate Runge-Kutta schemes. The scheme numerical capabilities at producing reliable accurate results are demonstrated by computing free-surface deformations of a shell in homogeneous compression and front deformations of a self-similar converging spherical shock wave. The interest of such a perturbation computation approach in hydrodynamic stability studies is examplified in the latter case by obtaining shock-front deformation dynamics results having no precedents with respect to accuracy and perturbation wavelength range

Faraday instability on a sphere: numerical simulation

We consider a spherical variant of the Faraday problem, in which a spherical drop is subjected to a time-periodic body force, as well as surface tension. We use a full three-dimensional parallel front-tracking code to calculate the interface motion of the parametrically forced oscillating viscous drop, as well as the velocity field inside and outside the drop. Forcing frequencies are chosen so as to excite spherical harmonic wavenumbers ranging from 1 to 6. We excite gravity waves for wavenumbers 1 and 2 and observe translational and oblate-prolate oscillation, respectively. For wavenumbers 3 to 6, we excite capillary waves and observe patterns analogous to the Platonic solids. For low viscosity, both subharmonic and harmonic responses are accessible. The patterns arising in each case are interpreted in the context of the theory of pattern formation with spherical symmetry

ORB5: a global electromagnetic gyrokinetic code using the PIC approach in toroidal geometry

This paper presents the current state of the global gyrokinetic code ORB5 as an update of the previous reference [Jolliet et al., Comp. Phys. Commun. 177 409 (2007)]. The ORB5 code solves the electromagnetic Vlasov-Maxwell system of equations using a PIC scheme and also includes collisions and strong flows. The code assumes multiple gyrokinetic ion species at all wavelengths for the polarization density and drift-kinetic electrons. Variants of the physical model can be selected for electrons such as assuming an adiabatic response or a ``hybrid'' model in which passing electrons are assumed adiabatic and trapped electrons are drift-kinetic. A Fourier filter as well as various control variates and noise reduction techniques enable simulations with good signal-to-noise ratios at a limited numerical cost. They are completed with different momentum and zonal flow-conserving heat sources allowing for temperature-gradient and flux-driven simulations. The code, which runs on both CPUs and GPUs, is well benchmarked against other similar codes and analytical predictions, and shows good scalability up to thousands of nodes

The effect of toroidal flows on the stability of ITGs in MAST

The free energy in the large temperature and density gradients in tokamaks can drive microinstabilities, which in turn drive turbulence. This turbulence is responsible for the transport of energy and particles over and above that predicted by neoclassical theory. Sheared toroidal rotation can suppress the turbulence and stabilise the underlying microinstabilities, thereby reducing the transport. This thesis investigates how variation of the equilibrium temperature and density profiles, over the same scales associated with the microinstabilities, affects how the ow shear stabilises the linear modes and suppresses the turbulence. A global gyrokinetic code is employed in this investigation, which retains the profile variation and simulates the full 3D domain of a tokamak plasma. How much ow shear is needed to stabilise the linear ion temperature gradient (ITG) mode is found to be dependent on its poloidal wavenumber, with longer wavelength modes needing more ow shear than the fastest growing mode. This dependence is present whether the ow shear is constant across the radius or if it has the variation typical in an experimental rotation profile. There is an asymmetry with respect to the sign of the ow shear in the effectiveness of the stabilisation, with the maximum linear growth rate occurring at finite negative shearing rates for the plasma studied here. This asymmetry arises from the profile variation, and is found to be significant in simulations of MAST L-mode plasmas, especially when the effects of kinetic trapped electrons are included in the simulations. Flow shear asymmetry is still present in nonlinear simulations, and the suppression of fully-developed turbulence depends on the sign of the shearing rate. With the experimental rotation profile, the heat ux arising from ITG turbulence is reduced by an amount comparable to the reduction in the linear growth rates. When the direction of the rotation profile is reversed, such that the sign of the ow shear is ipped while the magnitude remains the same, the turbulence is almost completely suppressed. A new diagnostic on MAST, beam emission spectroscopy (BES), is used to make a direct comparison between density fluctuations from simulation, and from experiment. Collisionless, electrostatic simulations with rotation are found to disagree significantly with experiment in the level of ITG turbulence activity and the correlation times and lengths of the turbulence. The inclusion of electron-electron and electron-ion collisions into static simulations is enough to bring the level of turbulent density uctuations down to within a factor two of the experimental levels, with the correlation lengths becoming comparable, while the correlation times remain an order of magnitude too large