Magnetohydrodynamic turbulence affects both terrestrial and astrophysical plasmas. The properties of magnetized turbulence must be better understood to more accurately characterize these systems. This work presents ideal MHD simulations of the compressible Taylor-Green vortex under a range of initial sub-sonic Mach numbers and magnetic field strengths. We find that regardless of the initial field strength, the magnetic energy becomes dominant over the kinetic energy on all scales after at most several dynamical times. The spectral indices of the kinetic and magnetic energy spectra become shallower than $k^{-5/3}$ over time and generally fluctuate. Using a shell-to-shell energy transfer analysis framework, we find that the magnetic fields facilitate a significant amount of the energy flux and that the kinetic energy cascade is suppressed. Moreover, we observe nonlocal energy transfer from the large scale kinetic energy to intermediate and small scale magnetic energy via magnetic tension. We conclude that even in intermittently or singularly driven weakly magnetized systems, the dynamical effects of magnetic fields cannot be neglected.
Rayleigh-Taylor instability in a compressible fluid is reconsidered. The density is allowed to vary with pressure under the barotropy assumption. For the case with equal speeds of sound in the two superposed fluids, in order to give a non-trivial compressibility correction to the Rayleigh-Taylor growth rate, the compressibility correction is calculated to $O(g^2/k^2a^4)$. To this order, compressibility effects are found to reduce the growth rate.
We present results from numerical simulations of Rayleigh-Taylor turbulence, performed using a recently proposed lattice Boltzmann method able to describe consistently a thermal compressible flow subject to an external forcing. The method allowed us to study the system both in the nearly-Boussinesq and strongly compressible regimes. Moreover, we show that when the stratification is important, the presence of the adiabatic gradient causes the arrest of the mixing process.
We investigate numerically a traveling wave pattern observed in experimental magnetized Taylor-Couette flow at low magnetic Reynolds number. By accurately modeling viscous and magnetic boundaries in all directions, we reproduce the experimentally measured wave patterns and their amplitudes. Contrary to previous claims, the waves are shown to be transiently amplified disturbances launched by viscous boundary layers rather than globally unstable magnetorotational modes.
We investigate the evolution of vortex-surface fields (VSFs) in compressible Taylor--Green flows at Mach numbers ($Ma$) ranging from 0.5 to 2.0 using direct numerical simulation. The formulation of VSFs in incompressible flows is extended to compressible flows, and a mass-based renormalization of VSFs is used to facilitate characterizing the evolution of a particular vortex surface. The effects of the Mach number on the VSF evolution are different in three stages. In the early stage, the jumps of the compressive velocity component near shocklets generate sinks to contract surrounding vortex su…
In this paper we present axisymmetric nonlinear simulations of magnetized Ekman and Stewartson layers in a magnetized Taylor-Couette flow with a centrifugally stable angular-momemtum profile and with a magnetic Reynolds number below the threshold of magnetorotational instability. The magnetic field is found to inhibit the Ekman suction. The width of the Ekman layer is reduced with increased magnetic field normal to the end plate. A uniformly-rotating region forms near the outer cylinder. A strong magnetic field leads to a steady Stewartson layer emanating from the junction between differential…
Taylor-Couette flow is often used as a simplified model for complex rotating flows in the interior of stars and accretion disks. The flow dynamics in these objects is influenced by magnetic fields. For example, quasi-Keplerian flows in Taylor-Couette geometry become unstable to a travelling or standing wave in an external magnetic field if the fluid is conducting; there is an instability even when the flow is hydrodynamically stable. This magnetorotational instability leads to the development of chaotic states and, eventually, turbulence, when the cylinder rotation is sufficiently fast. The tr…
The helical magnetorotational instability of the magnetized Taylor-Couette flow is studied numerically in a finite cylinder. A distant upstream insulating boundary is shown to stabilize the convective instability entirely while reducing the growth rate of the absolute instability. The reduction is less severe with larger height. After modeling the boundary conditions properly, the wave patterns observed in the experiment turn out to be a noise-sustained convective instability. After the source of the noise resulted from unstable Ekman and Stewartson layers is switched off, a slowly-decaying in…
Rayleigh-Taylor (RT) instability widely exists in nature and engineering fields. How to better understand the physical mechanism of RT instability is of great theoretical significance and practical value. At present, abundant results of RT instability have been obtained by traditional macroscopic methods. However, research on the thermodynamic non-equilibrium (TNE) effects in the process of system evolution is relatively scarce. In this paper, the discrete Boltzmann method based on non-equilibrium statistical physics is utilized to study the effects of the specific heat ratio on compressible R…
3-D extended-MHD simulations of the magnetized ablative Rayleigh-Taylor instability are presented for the first time. Previous 2-D simulations claiming perturbation suppression by magnetic tension are shown to be misleading, as they do not include the most unstable dimension. For perturbation modes along the applied field direction, the magnetic field simultaneously reduces ablative stabilization and adds magnetic tension stabilization; the stabilizing term is found to dominate for applied fields >5T, with both effects increasing in importance at short wavelengths. For modes perpendicular to t…
Scalar Dissipation Criticality in Compressible Magnetized Turbulence
ETH Zürich（苏黎世联邦理工学院）；Flatiron Institute, Simons Foundation（西蒙斯基金会平顿研究所）；School of Mathematics, IAS, Princeton（普林斯顿高等研究院数学学院）；GSSI, L’Aquila, Italy（意大利拉奎拉全球南方科学研究所）
Hollerbach and Rudiger have reported a new type of magnetorotational instability (MRI) in magnetized Taylor-Couette flow in the presence of combined axial and azimuthal magnetic fields. The salient advantage of this "helical'' MRI (HMRI) is that marginal instability occurs at arbitrarily low magnetic Reynolds and Lundquist numbers, suggesting that HMRI might be easier to realize than standard MRI (axial field only). We confirm their results, calculate HMRI growth rates, and show that in the resistive limit, HMRI is a weakly destabilized inertial oscillation propagating in a unique direction al…
The effects of isothermal stratification strength on vorticity dynamics for single-mode Rayleigh-Taylor instability (RTI) are examined using two dimensional fully compressible wavelet-based direct numerical simulations. The simulations model low Atwood number ($A=0.04$) RTI development for four different stratification strengths, corresponding to Mach numbers from 0.3 (weakly stratified) to 1.2 (strongly stratified), and for three different perturbation Reynolds numbers, from 5,000 to 20,000. All simulations use adaptive wavelet-based mesh refinement to achieve very fine spatial resolutions at…
The linearized Navier-Stokes equations for a system of superposed immiscible compressible ideal fluids are analyzed. The results of the analysis reconcile the stabilizing and destabilizing effects of compressibility reported in the literature. It is shown that the growth rate $n$ obtained for an inviscid, compressible flow in an infinite domain is bounded by the growth rates obtained for the corresponding incompressible flows with uniform and exponentially varying density. As the equilibrium pressure at the interface $p_\infty$ increases (less compressible flow), $n$ increases towards the unif…
A conducting Taylor-Couette flow with quasi-Keplerian rotation law containing a toroidal magnetic field serves as a mean-field dynamo model of the Tayler-Spruit-type. The flows are unstable against nonaxisymmetric perturbations which form electromotive forces defining $α$ effect and eddy diffusivity. If both degenerated modes with $m=\pm 1$ are excited with the same power then the global $α$ effect vanishes and a dynamo cannot work. It is shown, however, that the Tayler instability produces finite $α$ effects if only an isolated mode is considered but this intrinsic helicity of the single-mode…
Rayleigh-Taylor instability (RTI) occurs at the interface of two media when the heavier fluid is accelerated into the lighter fluid and is a prototypical hydrodynamic event present in many physical events. In high energy physics, this manifests itself across a wide range of length scales from nuclear confinement fusion at micron-scale to supernova explosion at terra scales. RTI can also be viewed as a baroclinic instability prevalent in engineering, geophysics, and astrophysics, a pedagogic description of which is given in Sengupta {\it et al., Comput. Fluids,} {\bf 225}, 104995 (2021) with re…
The variational principle for linear stability of three-dimensional, inhomogenious, compressible, moving magnetized plasma is suggested. The principle is ``softer'' (easier to be satisfied) than all previously known variational stability conditions. The key point of the analysis is a conservation in variations of new integrals inherent in the linearized equation of the motion that was not earlier discussed in the literature.
We develop a novel multiscale model of interface motion for the Rayleigh-Taylor instability (RTI) and Richtmyer-Meshkov instability (RMI) for two-dimensional, inviscid, compressible flows with vorticity, which yields a fast-running numerical algorithm that produces both qualitatively and quantitatively similar results to a resolved gas dynamics code, while running approximately two orders of magnitude (in time) faster. Our multiscale model is founded upon a new compressible-incompressible decomposition of the velocity field $u=v+w$. The incompressible component $w$ of the velocity is also irro…
Magnetorotational instability (MRI) is one of the fundamental processes in astrophysics, driving angular momentum transport and mass accretion in a wide variety of cosmic objects. Despite much theoretical/numerical and experimental efforts over the last decades, its saturation mechanism and amplitude, which sets the angular momentum transport rate, remains not well understood, especially in the limit of high resistivity, or small magnetic Prandtl numbers typical to interiors (dead zones) of protoplanetary disks, liquid cores of planets and liquid metals in laboratory. Using direct numerical si…
Off-lattice Boltzmann methods increase the flexibility and applicability of lattice Boltzmann methods by decoupling the discretizations of time, space, and particle velocities. However, the velocity sets that are mostly used in off-lattice Boltzmann simulations were originally tailored to on-lattice Boltzmann methods. In this contribution, we show how the accuracy and efficiency of weakly and fully compressible semi-Lagrangian off-lattice Boltzmann simulations is increased by velocity sets derived from cubature rules, i.e. multivariate quadratures, which have not been produced by the Gauss-pro…
Vectorial discrete unified gas kinetic scheme for continuum compressible flows
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；State Key Laboratory of Coal Combustion, School of Energy and Power Engineering, Huazhong University of Science and Technology（华中科技大学能源与动力工程学院煤炭燃烧国家重点实验室）；Institute of Interdisciplinary Research for Mathematics and Applied Science, Huazhong University of Science and Technology（华中科技大学数学与应用科学交叉研究院）
A conservative Godunov-type mesh-free hydrodynamics scheme for weakly compressible multiphase flows
Technical University of Munich (TUM)（慕尼黑工业大学）；Munich Institute of Integrated Materials, Energy and Process Engineering (MEP), Technical University of Munich (TUM)（慕尼黑工业大学集成材料、能源和工艺工程研究所）
Supersonic turbulent flows of magnetized gas are believed to play an important role in the dynamics of star-forming clouds in galaxies. Understanding statistical properties of such flows is crucial for developing a theory of star formation. In this letter we propose a unified approach for obtaining the velocity scaling in compressible and super--Alfvénic turbulence, valid for arbitrary sonic Mach number, \ms. We demonstrate with numerical simulations that the scaling can be described with the She--Lévêque formalism, where only one parameter, interpreted as the Hausdorff dimension of the most i…
We explore the suitability of deep learning to capture the physics of subgrid-scale ideal magnetohydrodynamics turbulence of 2-D simulations of the magnetized Kelvin-Helmholtz instability. We produce simulations at different resolutions to systematically quantify the performance of neural network models to reproduce the physics of these complex simulations. We compare the performance of our neural networks with gradient models, which are extensively used in the extensively in the magnetohydrodynamic literature. Our findings indicate that neural networks significantly outperform gradient models…
An immersed-boundary (IB) method is proposed and applied in the gas-kinetic BGK scheme to simulate incompressible/compressible viscous flow with stationary/moving boundary. In present method the ghost-cell technique is adopted to fulfill the boundary condition on the immersed boundary. A novel idea "local boundary determination" is put forward to identify the ghost cells, each of which may have several different ghost-cell constructions corresponding to different boundary segments, thus eliminating the singularity of the ghost cell. Furthermore, the so-called "fresh-cell" problem when implemen…
Turbulent compressible flows are traditionally simulated using explicit time integrators applied to discretized versions of the Navier-Stokes equations. However, the associated Courant-Friedrichs-Lewy condition severely restricts the maximum time step size. Exploiting the Lagrangian nature of the Boltzmann equation's material derivative, we now introduce a feasible three-dimensional semi-Lagrangian lattice Boltzmann method (SLLBM), which circumvents this restriction. While many lattice Boltzmann methods for compressible flows were restricted to two dimensions due to the enormous number of disc…
We report a detailed study of Eulerian and Lagrangian statistics from high resolution Direct Numerical Simulations of isotropic weakly compressible turbulence. Reynolds number at the Taylor microscale is estimated to be around 600. Eulerian and Lagrangian statistics is evaluated over a huge data set, made by $1856^3$ spatial collocation points and by 16 million particles, followed for about one large-scale eddy turn over time. We present data for Eulerian and Lagrangian Structure functions up to ten order. We analyse the local scaling properties in the inertial range and in the viscous range. …
Nonlinear evolution of magnetorotational instability in a magnetized Taylor-Couette flow: scaling properties and relation to upcoming DRESDYN-MRI experiment
A systematic analysis of the discrete conservation properties of non-dissipative, central-difference approximations of the compressible Navier-Stokes equations is reported. A general triple splitting of the nonlinear convective terms is considered, and energy-preserving formulations are fully characterized by deriving a two-parameter family of split forms. Previously developed formulations reported in literature are shown to be particular members of this family; novel splittings are introduced and discussed as well. Furthermore, the conservation properties yielded by different choices for the …
It is shown that the universal steady Euler flow field, independent of boundary shape or symmetry, in a toroidal domain with fixed boundary obeys a nonlinear Beltrami equation, with the nonlinearity arising from a Boltzmann-like, velocity-dependent factor. Moreover, this is a relaxed velocity field, in the sense that it extremizes the total kinetic energy in the domain under free variations of the velocity field, constrained only by tangential velocity and vorticity boundary conditions and conservation of total fluid helicity and entropy. This is analogous to Woltjer-Taylor relaxation of plasm…
This work thoroughly investigates a semi-Lagrangian lattice Boltzmann (SLLBM) solver for compressible flows. In contrast to other LBM for compressible flows, the vertices are organized in cells, and interpolation polynomials up to fourth order are used to attain the off-vertex distribution function values. Differing from the recently introduced Particles on Demand (PoD) method, the method operates in a static, non-moving reference frame. Yet the SLLBM in the present formulation grants supersonic flows and exhibits a high degree of Galilean invariance. The SLLBM solver allows for an independent…
The Rayleigh-Taylor Instability (RTI) under multi-mode perturbation in compressible flow is probed via the Discrete Boltzmann Modeling (DBM) with tracers. The distribution of tracers provides clear boundaries between light and heavy fluids in the position space. Besides, the position-velocity phase space offers a new perspective for understanding the flow behavior of RTI with intuitive geometrical correspondence. The effects of viscosity, acceleration, compressibility, and Atwood number on the mixing of material and momentum and the mean non-equilibrium strength at the interfaces are investiga…
Accurate numerical modeling of compressible flows, particularly in the turbulent regime, requires a method that is non-dissipative and stable at high Reynolds ($Re$) numbers. For a compressible flow, it is known that discrete conservation of kinetic energy is not a sufficient condition for numerical stability, unlike in incompressible flows. In this study, we adopt the recently developed conservative diffuse-interface method (Jain, Mani $\&$ Moin, $\textit{J. Comput. Phys.}$, 2020) along with the five-equation model for the simulation of compressible two-phase flows. This method discretely con…
Performing accurate large eddy simulations in compressible, turbulent magnetohydrodynamics is more challenging than in non-magnetized fluids due to the complex interplay between kinetic, magnetic and internal energy at different scales. Here we extend the sub-grid-scale gradient model, so far used in the momentum and induction equations, to account also for the unresolved scales in the energy evolution equation of a compressible ideal MHD fluid with a generic equation of state. We assess the model by considering box simulations of the turbulence triggered across a shear layer by the Kelvin-Hel…
In this paper, the entropy conservative/stable algorithms presented by Del Rey Fernandez and coauthors [18,16,17] for the compressible Euler and Navier-Stokes equations on nonconforming p-refined/coarsened curvilinear grids is extended to h/p refinement/coarsening. The main difficulty in developing nonconforming algorithms is the construction of appropriate coupling procedures across nonconforming interfaces. Here, a computationally simple and efficient approach based upon using decoupled interpolation operators is utilized. The resulting scheme is entropy conservative/stable and element-wise …
