Application of Molecular Hydrodynamics to Astrophysical Flows. II: Unconditional Stability (original) (raw)

Application of the Molecular Hydrodynamics to Accretion Flows

We propose a new CFD method modified from the Direct Simulation Monte Carlo method (DSMC). The DSMC is modified to handle continuum flows by setting the mean free path of molecules to a fixed, constant value, which is on the order of a cell size ∆x. By doing so, we quickly reach equilibrium. We call our scheme 'Molecular hydrodynamics' (MH). We test our MH scheme with three typical examples: the Sod shock tube problem, a two-dimensional step flow and a two-dimensional accretion disc. The MH has the following characteristics: 1) Time steps are not restricted by a CFL condition in spite of the fact that the scheme is time explicit; 2) The quality of the solutions is improved by an ensemble averaging.

Grid-Based Hydrodynamics in Astrophysical Fluid Flows

In this review, the equations of hydrodynamics, magnetohydrodynamics, and radiation hydrodynamics are presented, together with their corresponding nonideal source terms. I overview the current landscape of modern grid-based numerical techniques with an emphasis on numerical diffusion, which plays a fundamental role in stabilizing the solution but is also the main source of errors associated with these numerical techniques. I discuss in great detail the inclusion of additional important source terms, such as cooling and gravity. I also show how to modify classic operator-splitting techniques to avoid undesirable numerical errors associated with these additional source terms, in particular in the presence of highly supersonic flows. I finally present various mesh adaptation strategies that can be used to minimize these numerical errors. To conclude, I review existing astrophysical software that is publicly available to perform simulations for such astrophysical fluids.

The Physics of Fluids and Plasmas: An Introduction for Astrophysicists

Nuclear Fusion, 1999

Preface xiii Acknowledgements xvii 2 Boltzmann equation 2.1 Collisions in a dilute neutral gas 19 2.2 The collision integral 2.3 The Maxwellian distribution 2.4 Boltzmann's H theorem 2.5 The conservation equation Exercises 3 March towards hydrodynamics 3.1 The moment equations 3.2 An excursion into stellar dynamics. Oort limit 3.3 Zero-order approximation 3.4 Transport phenomena 3.5 Comparison with experiments 3.6 Hydrodynamics at last 3.7 Concluding remarks Exercises vn viii Contents Properties of ideal fluids 4.1 Macroscopic derivation of hydrodynamic equations 4.1.1 The equation of motion 4.1.2 The equation of energy 4.2 The vorticity equation. Incompressible and barotropic fluids 4.3 Hydrodynamic equations in conservative forms 4.4 Hydrostatics. Modelling the solar corona 4.5 Bernoulli's principle for steady flows 4.6 Kelvin's vorticity theorem 4.7 Potential flows. Flow past a cylinder 4.8 Stream function Exercises Viscous flows 5.1 Tangential stress in a Newtonian fluid 5.2 Navier-Stokes equation 5.3 Flow through a circular pipe 5.4 Scaling and Reynolds number 5.5 Viscous flow past solid bodies. Boundary layers 5.6 Aerodynamic lift 5.7 Accretion disks in astrophysics 5.7.1 The basic disk dynamics 5.7.2 Steady disk Exercises Gas dynamics 6.1 Thermodynamic properties of a perfect gas 6.2 Acoustic waves 6.3 Emission of acoustic waves 6.4 Steepening into shock waves. The method of characteristics 6.5 The structure of shock waves 6.6 Spherical blast waves. Supernova explosions 6.7 One-dimensional gas flow. Extragalactic jets 6.8 Spherical accretion and winds Exercises Linear theory of waves and instabilities 7.1 The philosophy of perturbation analysis 7.2 Convective instability and internal gravity waves 7.3 Rayleigh-Benard convection 7.4 Perturbations at a two-fluid interface 7.4.1 Surface gravity waves 7.4.2 Rayleigh-Taylor instability

The Physics of Fluids and Plasmas: An Introduction for Astrophysicists. Edited by A. R. CHOUDHURI. Cambridge University Press, 1998. 448 pp. ISBN 0521 55487 X, £52.50 (hardback); ISBN 0521 55543 4, £19.95 (paperback)

Journal of Fluid Mechanics, 1999

Preface xiii Acknowledgements xvii 2 Boltzmann equation 2.1 Collisions in a dilute neutral gas 19 2.2 The collision integral 2.3 The Maxwellian distribution 2.4 Boltzmann's H theorem 2.5 The conservation equation Exercises 3 March towards hydrodynamics 3.1 The moment equations 3.2 An excursion into stellar dynamics. Oort limit 3.3 Zero-order approximation 3.4 Transport phenomena 3.5 Comparison with experiments 3.6 Hydrodynamics at last 3.7 Concluding remarks Exercises vn viii Contents Properties of ideal fluids 4.1 Macroscopic derivation of hydrodynamic equations 4.1.1 The equation of motion 4.1.2 The equation of energy 4.2 The vorticity equation. Incompressible and barotropic fluids 4.3 Hydrodynamic equations in conservative forms 4.4 Hydrostatics. Modelling the solar corona 4.5 Bernoulli's principle for steady flows 4.6 Kelvin's vorticity theorem 4.7 Potential flows. Flow past a cylinder 4.8 Stream function Exercises Viscous flows 5.1 Tangential stress in a Newtonian fluid 5.2 Navier-Stokes equation 5.3 Flow through a circular pipe 5.4 Scaling and Reynolds number 5.5 Viscous flow past solid bodies. Boundary layers 5.6 Aerodynamic lift 5.7 Accretion disks in astrophysics 5.7.1 The basic disk dynamics 5.7.2 Steady disk Exercises Gas dynamics 6.1 Thermodynamic properties of a perfect gas 6.2 Acoustic waves 6.3 Emission of acoustic waves 6.4 Steepening into shock waves. The method of characteristics 6.5 The structure of shock waves 6.6 Spherical blast waves. Supernova explosions 6.7 One-dimensional gas flow. Extragalactic jets 6.8 Spherical accretion and winds Exercises Linear theory of waves and instabilities 7.1 The philosophy of perturbation analysis 7.2 Convective instability and internal gravity waves 7.3 Rayleigh-Benard convection 7.4 Perturbations at a two-fluid interface 7.4.1 Surface gravity waves 7.4.2 Rayleigh-Taylor instability

Equations and Algorithms for Astrophysical Radiation Hydrodynamics in All Non-Relativistic Regimes

2006

We analyze the equations of radiation hydrodynamics under the approximations of flux-limited diffusion and a thermal radiation field, and derive the minimal set of evolution equations that includes all terms that are of leading order in any regime of non-relativistic radiation hydrodynamics. Our equations are accurate to first order in v/c in the static diffusion regime. We give the equations in a conservation law form well-suited to implementation in numerical algorithms. Our work improves on previous zeroth order derivations of the equations by retaining differences between laboratory frame and comoving frame quantities, which are neglected at zeroth order. We compare our equations to the zeroth order equations, and show that in certain regimes the zeroth order equations omit terms that are formally of leading order. We discuss the circumstances under which this will produce significant errors. For systems in the static diffusion regime, our analysis of the equations suggests an a...

Following the flow: tracer particles in astrophysical fluid simulations

Monthly Notices of the Royal Astronomical Society, 2013

We present two numerical schemes for passive tracer particles in the hydrodynamical movingmesh code AREPO, and compare their performance for various problems, from simple set-ups to cosmological simulations. The purpose of tracer particles is to allow the flow to be followed in a Lagrangian way, tracing the evolution of the fluid with time, and allowing the thermodynamical history of individual fluid parcels to be recorded. We find that the commonly used 'velocity field tracers', which are advected using the fluid velocity field, do not in general follow the mass flow correctly, and explain why this is the case. This method can result in orderof-magnitude biases in simulations of driven turbulence and in cosmological simulations, rendering the velocity field tracers inappropriate for following these flows. We then discuss a novel implementation of 'Monte Carlo tracers', which are moved along with fluid cells and are exchanged probabilistically between them following the mass flux. This method reproduces the mass distribution of the fluid correctly. The main limitation of this approach is that it is more diffusive than the fluid itself. Nonetheless, we show that this novel approach is more reliable than that has been employed previously and demonstrate that it is appropriate for following hydrodynamical flows in mesh-based codes. The Monte Carlo tracers can also naturally be transferred between fluid cells and other types of particles, such as stellar particles, so that the mass flow in cosmological simulations can be followed in its entirety.

Testing the concept of integral approach to derivatives within the smoothed particle hydrodynamics technique in astrophysical scenarios

The behavior of IAD_0 scheme, a fully conservative SPH scheme based on a tensor formulation, is analyzed in connection with several astrophysical scenarios, and compared to the same simulations carried out with the standard SPH technique. The proposed hydrodynamic scheme is validated using a variety of numerical tests that cover important topics in astrophysics, such as the evolution of supernova remnants, the stability of self-gravitating bodies and the coalescence of compact objects. The results suggest that the SPH scheme built with the integral approach to the derivatives premise improves the results of the standard SPH technique. In particular, it is observed a better development of hydrodynamic instabilities, an improved description of self-gravitant structures in equilibrium and a reasonable description of the process of coalescence of two white dwarfs. A good energy, and linear and angular momentum conservation, generally better than that of standard SPH, was also obtained. In addition the new scheme is less susceptible to suffer pairing instability.

Direct Simulation Monte Carlo for astrophysical flows - I. Motivation and methodology

Monthly Notices of the Royal Astronomical Society, 2014

We describe a hybrid Direct Simulation Monte Carlo (DSMC) code for simultaneously solving the collisional Boltzmann equation for gas and the collisionless Boltzmann equation for stars and dark matter for problems important to galaxy evolution. This project is motivated by the need to understand the controlling dynamics at interfaces between gases of widely differing densities and temperature, i.e. multiphase media. While more expensive than hydrodynamics, the kinetic approach does not suffer from discontinuities and it applies when the continuum limit does not, such as in the collapse of galaxy clusters and at the interface between coronal halo gas and a thin neutral gas layer. Finally, the momentum flux is carried, self-consistently, by particles and this approach explicitly resolves and thereby 'captures' shocks. The DSMC method splits the solution into two pieces: 1) the evolution of the phase-space flow without collisions; and 2) the evolution governed the collision term alone without phase-space flow. This splitting approach makes DSMC an ideal match to existing particle-based n-body codes. If the mean free path becomes very small compared to any scale of interest, the method abandons simulated particle collisions and simply adopts the relaxed solution in each interaction cell consistent with the overall energy and momentum fluxes. This is functionally equivalent to solving the Navier-Stokes equations on a mesh. Our implementation is tested using the Sod shock tube problem and the non-linear development of an Kelvin-Helmholtz unstable shear layer.