Role of critical points of the skin friction field in formation of plumes in thermal convection

The dynamics in the thin boundary layers of temperature and velocity is the key to a deeper understanding of turbulent transport of heat and momentum in thermal convection. The velocity gradient at the hot and cold plates of a Rayleigh-B\'{e}nard convection cell forms the two-dimensional skin friction field and is related to the formation of thermal plumes in the respective boundary layers. Our analysis is based on a direct numerical simulation of Rayleigh-B\'{e}nard convection in a closed cylindrical cell of aspect ratio $\Gamma=1$ and focused on the critical points of the skin friction field. We identify triplets of critical points, which are composed of two unstable nodes and a saddle between them, as the characteristic building block of the skin friction field. Isolated triplets as well as networks of triplets are detected. The majority of the ridges of line-like thermal plumes coincide with the unstable manifolds of the saddles. From a dynamical Lagrangian perspective, thermal plumes are formed together with an attractive hyperbolic Lagrangian Coherent Structure of the skin friction field. We also discuss the differences from the skin friction field in turbulent channel flows from the perspective of the Poincar\'{e}-Hopf index theorem for two-dimensional vector fields

Magnetohydrodynamic duct and channel flows at finite magnetic Reynolds numbers

Magnetohydrodynamic duct flows have so far been studied only in the limit of negligible magnetic Reynolds numbers ($R_m$). When $R_m$ is finite, the secondary magnetic field becomes significant, leading to a fully coupled evolution of the magnetic field and the conducting flow. Characterization of such flows is essential in understanding wall-bounded magnetohydrodynamic turbulence at finite $R_m$ as well as in industrial applications like the design of electromagnetic pumps and measurement of transient flows using techniques such as Lorentz force velocimetry. This thesis presents the development of a numerical framework for direct numerical simulations (DNS) of magnetohydrodynamic flows in straight rectagular ducts at finite $R_m$, which is subsequently used to study three specific problems. The thesis opens with a brief overview of MHD and a review of the existing state of art in duct and channel MHD flows. This is followed by a description of the physical model governing the problem of MHD duct flow with insulating walls and streamwise periodicity. In the main part of the thesis, a hybrid finite difference-boundary element computational procedure is developed that is used to solve the magnetic induction equation with boundary conditions that satisfy interior-exterior matching of the magnetic field at the domain wall boundaries. The numerical procedure is implemented into a code and a detailed verification of the same is performed in the limit of low $R_m$ by comparing with the results obtained using a quasistatic approach that has no coupling with the exterior. Following this, the effect of $R_m$ on the transient response of Lorentz force is studied using the problem of a strongly accelerated solid conducting bar in the presence of an imposed localized magnetic field. The response time of Lorentz force depends linearly on $R_m$ and shows a good agreement with the existing experiments. For sufficiently large values of $R_m$, the peak Lorentz force is found to show an ${R_m}^{-1}$ dependence. After this, the phenomenon of dynamic runaway due to magnetic flux expulsion in a two-dimensional channel flow is studied. Comparison with an existing one-dimensional model shows a close agreement for the Hartmann regime and the bifurcation location but the model overpredicts the core velocities in the Poisuelle regime significantly. Parametric studies indicate the importance of the streamwise wavenumber of the imposed magnetic field on the bifurcation point. Finally, turbulent Hartmann duct flow is investigated at moderate vaues of $R_m$. A higher $R_m$ is found to delay the onset of relaminarization to a higher value of Hartmann number. Large scale turbulence is induced at moderate $R_m$ and the effect increases with $R_m$. Between the core and the Shercliff layers, Reynolds stresses decrease with increase in $R_m$, leading to larger mean velocities in that region.Magnetohydrodynamische Kanalströmungen (MHD-KS) wurden bisher nur bei vernachlässigbar kleiner magnetischer Reynoldszahl $R_m$ untersucht. Bei endlichem $R_m$ wird das sekundäre Magnetfeld signifikant, was zu einer gekoppelten Entwicklung von Magnetfeld und leitfähiger Strömung führt. Die Charakterisierung solcher Strömungen ist essentiell für das Verständnis von wandbegrenzter MHD-Turbulenz und in Anwendungen wie z.B. elektromagnetischen Pumpen und der induktiven Strömungsmessung. Die Dissertation stellt ein Verfahren für die direkte numerische Simulation (DNS) von MHD-KS bei endlichem $R_m$ vor, welches dann auf drei Probleme angewendet wird. Am Anfang der Arbeit steht eine kurze Übersicht zur MHD und zum Stand des Wissens zu MHD-KS. Danach folgt eine Beschreibung des physikalischen Modells für die MHD-KS mit elektrisch isolierenden Wänden. Im Hauptteil der Arbeit wird ein hybrides Berechnungsverfahren entwickelt und implementiert, das auf finiten Differenzen sowie dem Randintegralverfahren basiert. Es dient zur Lösung der Induktionsgleichung mit Randbedingungen, die für einen stetigen Anschluss des Magnetfelds auf den Gebietsrändern zwischen Innen- und Außenraum sorgen. Eine detaillierte Verifikation des Codes wird durch Vergleich mit der quasistatischen Näherung vorgenommen. Anschliessend wird das Zeitverhalten der Lorentzkraft bei beschleunigter Bewegung einer leitfähigen rechteckigen Stange in einem lokalisierten Magnetfeld untersucht. Die Zeitantwort der Lorentzkraft hängt linear von $R_m$ ab und stimmt gut mit Experimenten überein. Für große $R_m$ sind die Maximalwerte der Lorentzkraft umgekehrt proportional zu $R_m$. Im weiteren wird das dynamische ``Weglaufen'' der Geschwindigkeit infolge von magnetischer Flussverdrängung in einer zweidimensionalen MHD-KS untersucht. Der Vergleich mit einem eindimensionalen Modell zeigt eine gute Übereinstimmung für das sogenannte Hartmann-Regime und den Bifurkationspunkt zum sogenannten Poiseuille-Regime, bei dem allerdings die Geschwindigkeit vom Modell überschätzt wird. Die Wellenlänge des Magnetfelds ist für den Bifurkationspunkt entscheidend. Abschliessend wird die turbulente Hartmannströmung untersucht. Bei endlichem $R_m$ verschiebt sich die Relaminarisierung zu größeren Hartmannzahlen und es wird großsk-alige Turbulenz angeregt. Zwischen den Shercliff-Schichten und dem Strömungskern verringern sich die Reynoldsspannungen mit steigendem $R_m$, was zu höherer mittlerer Geschwindigkeit und flacheren Geschwindigkeitsprofilen führt

The JOREK non-linear extended MHD code and applications to large-scale instabilities and their control in magnetically confined fusion plasmas

JOREK is a massively parallel fully implicit non-linear extended magneto-hydrodynamic (MHD) code for realistic tokamak X-point plasmas. It has become a widely used versatile simulation code for studying large-scale plasma instabilities and their control and is continuously developed in an international community with strong involvements in the European fusion research programme and ITER organization. This article gives a comprehensive overview of the physics models implemented, numerical methods applied for solving the equations and physics studies performed with the code. A dedicated section highlights some of the verification work done for the code. A hierarchy of different physics models is available including a free boundary and resistive wall extension and hybrid kinetic-fluid models. The code allows for flux-surface aligned iso-parametric finite element grids in single and double X-point plasmas which can be extended to the true physical walls and uses a robust fully implicit time stepping. Particular focus is laid on plasma edge and scrape-off layer (SOL) physics as well as disruption related phenomena. Among the key results obtained with JOREK regarding plasma edge and SOL, are deep insights into the dynamics of edge localized modes (ELMs), ELM cycles, and ELM control by resonant magnetic perturbations, pellet injection, as well as by vertical magnetic kicks. Also ELM free regimes, detachment physics, the generation and transport of impurities during an ELM, and electrostatic turbulence in the pedestal region are investigated. Regarding disruptions, the focus is on the dynamics of the thermal quench (TQ) and current quench triggered by massive gas injection and shattered pellet injection, runaway electron (RE) dynamics as well as the RE interaction with MHD modes, and vertical displacement events. Also the seeding and suppression of tearing modes (TMs), the dynamics of naturally occurring TQs triggered by locked modes, and radiative collapses are being studied.Peer ReviewedPostprint (published version

Turbulent magnetohydrodynamic flow in a square duct: Comparison of zero and finite magnetic Reynolds number cases

Three-dimensional turbulent magnetohydrodynamic flow in a duct with a square cross section and insulating walls is investigated by direct numerical simulations. The flow evolves in the presence of a uniform vertical magnetic field and is driven by an applied mean pressure gradient. A boundary element technique is applied to treat the magnetic field boundary conditions at the walls consistently. Our primary focus is on the large- and small-scale characteristics of turbulence in the regime of moderate magnetic Reynolds numbers up to Rm $∼10^2$ and a comparison of the simulations with the quasistatic limit at Rm=0. The present simulations demonstrate that differences to the quasistatic case arise for the accessible magnetic Prandtl number Pm $∼10^{−2}$ and different Hartmann numbers up to Ha=43.5. Hartmann and Shercliff layers at the duct walls are affected differently when a dynamical coupling to secondary magnetic fields is present. This becomes manifest by the comparison of the mean streamwise velocity profiles as well as the skin friction coefficients. While large-scale properties change only moderately, the impact on small-scale statistics is much stronger as quantified by an analysis of local anisotropy based on velocity derivatives. The small-scale anisotropy is found to increase at moderate Rm. These differences can be attributed to the additional physical phenomena which are present when secondary magnetic fields evolve, such as the expulsion of magnetic flux in the bulk of the duct or the presence of turbulent electromotive forces

A hybrid finite difference–boundary element procedure for the simulation of turbulent MHD duct flow at finite magnetic Reynolds number

A conservative coupled finite difference-boundary element computational procedure for the simulation of turbulent magnetohydrodynamic flow in a straight rectangular duct at finite magnetic Reynolds number is presented. The flow is assumed to be periodic in the streamwise direction and is driven by a mean pressure gradient. The duct walls are considered to be electrically insulating. The co-evolution of the velocity and magnetic fields as described respectively by the Navier-Stokes and the magnetic induction equations, together with the coupling of the magnetic field between the conducting domain and the non-conducting exterior is solved using the magnetic field formulation. The aim is to simulate localized magnetic fields interacting with turbulent duct flow. Detailed verification of the implementation of the numerical scheme is conducted in the limiting case of low magnetic Reynolds number by comparing with the results obtained using a quasistatic approach that has no coupling with the exterior. The rigorous procedure with non-local magnetic boundary conditions is compared versus simplified pseudo-vacuum boundary conditions and the differences are quantified. Our first direct numerical simulations of turbulent Hartmann duct flow at moderate magnetic Reynolds numbers and a low flow Reynolds number show significant differences in the duct flow turbulence, even at low interaction level between the flow and magnetic fiel

The JOREK non-linear extended MHD code and applications to large-scale instabilities and their control in magnetically confined fusion plasmas

International audienceJOREK is a massively parallel fully implicit non-linear extended magneto-hydrodynamic(MHD) code for realistic tokamak X-point plasmas. It has become a widely used versatile simulation code for studying large-scale plasma instabilities and their control and is continuously developed in an international community with strong involvements in the European fusion research programme and ITER organization. This article gives a comprehensive overview of the physics models implemented, numerical methods applied for solving the equations and physics studies performed with the code. A dedicated section highlights some of the verification work done for the code. A hierarchy of different physics models is available including a free boundary and resistive wall extension and hybridkinetic-fluid models. The code allows for flux-surface aligned iso-parametric finite element grids in single and double X-point plasmas which can be extended to the true physical walls and uses a robust fully implicit time stepping. Particular focus is laid on plasma edge and scrape-off layer (SOL) physics as well as disruption related phenomena. Among the key results obtained with JOREK regarding plasma edge and SOL, are deep insights into the dynamics of edge localized modes (ELMs), ELM cycles, and ELM control by resonant magnetic perturbations, pellet injection, as well as by vertical magnetic kicks. Also ELM free regimes, detachment physics, the generation and transport of impurities during an ELM, and electrostatic turbulence in the pedestal region are investigated. Regarding disruptions, the focus is on the dynamics of the thermal quench (TQ) and current quench triggered by massive gas injection and shattered pellet injection, runaway electron (RE) dynamics as well as the RE interaction with MHD modes, and vertical displacement events. Also the seeding and suppression of tearing modes (TMs), the dynamics of naturally occurring TQs triggered by locked modes, and radiative collapses are being studied