Modelling shear flows with smoothed particle hydrodynamics and grid-based methods (original) (raw)
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Modelling Shear Flows with SPH and Grid Based Methods
Arxiv preprint arXiv: …, 2010
Given the importance of shear flows for astrophysical gas dynamics, we study the evolution of the Kelvin-Helmholtz instability (KHI) analytically and numerically. We derive the dispersion relation for the two-dimensional KHI including viscous dissipation. The resulting expression for the growth rate is then used to estimate the intrinsic viscosity of four numerical schemes depending on code-specific as well as on physical parameters. Our set of numerical schemes includes the Tree-SPH code VINE, an alternative SPH formulation developed by Price (2008), and the finite-volume grid codes FLASH and PLUTO. In the first part, we explicitly demonstrate the effect of dissipation-inhibiting mechanisms such as the Balsara viscosity on the evolution of the KHI. With VINE, increasing density contrasts lead to a continuously increasing suppression of the KHI (with complete suppression from a contrast of 6:1 or higher). The alternative SPH formulation including an artificial thermal conductivity reproduces the analytically expected growth rates up to a density contrast of 10:1. The second part addresses the shear flow evolution with FLASH and PLUTO. Both codes result in a consistent non-viscous evolution (in the equal as well as in the different density case) in agreement with the analytical prediction. The viscous evolution studied with FLASH shows minor deviations from the analytical prediction.
A Pure Hydrodynamic Instability in Shear Flows and Its Application to Astrophysical Accretion Disks
The Astrophysical Journal, 2016
We provide a possible resolution for the century-old problem of hydrodynamic shear flows, which are apparently stable in linear analysis but shown to be turbulent in astrophysically observed data and experiments. This mismatch is noticed in a variety of systems, from laboratory to astrophysical flows. There are so many uncountable attempts made so far to resolve this mismatch, beginning with the early work of Kelvin, Rayleigh, and Reynolds toward the end of the nineteenth century. Here we show that the presence of stochastic noise, whose inevitable presence should not be neglected in the stability analysis of shear flows, leads to pure hydrodynamic linear instability therein. This explains the origin of turbulence, which has been observed/interpreted in astrophysical accretion disks, laboratory experiments, and direct numerical simulations. This is, to the best of our knowledge, the first solution to the long-standing problem of hydrodynamic instability of Rayleigh-stable flows.
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.
Simulations of Astrophysical fluid instabilities
arXiv preprint astro-ph/0102239, 2001
Abstract: We present direct numerical simulations of mixing at Rayleigh-Taylor unstable interfaces performed with the FLASH code, developed at the ASCI/Alliances Center for Astrophysical Thermonuclear Flashes at the University of Chicago. We present initial results of single-mode studies in two and three dimensions. Our results indicate that three-dimensional instabilities grow significantly faster than two-dimensional instabilities and that grid resolution can have a significant effect on instability growth rates. We also find that ...
Kelvin-Helmholtz instabilities in smoothed particle hydrodynamics
Monthly Notices of the Royal Astronomical Society, 2010
In this paper we investigate whether Smoothed Particle Hydrodynamics (SPH), equipped with artificial conductivity, is able to capture the physics of density/energy discontinuities in the case of the so-called shearing layers test, a test for examining Kelvin-Helmholtz (KH) instabilities. We can trace back each failure of SPH to show KH rolls to two causes: i) shock waves travelling in the simulation box and ii) particle clumping, or more generally, particle noise. The probable cause of shock waves is the Local Mixing Instability (LMI), previously identified in the literature. Particle noise on the other hand is a problem because it introduces a large error in the SPH momentum equation. The shocks are hard to avoid in SPH simulations with initial density gradients because the most straightforward way of removing them, i.e. relaxing the initial conditions, is not viable. Indeed, by the time sufficient relaxing has taken place the density and energy gradients have become prohibitively wide. The particle disorder introduced by the relaxation is also a problem. We show that setting up initial conditions with a suitably smoothed density gradient dramatically improves results: shock waves are reduced whilst retaining relatively sharp gradients and avoiding unnecessary particle disorder. Particle clumping is easy to overcome, the most straightforward method being the use of a suitable smoothing kernel with non-zero first central derivative. We present results to that effect using a new smoothing kernel: the LInear Quartic (LIQ) kernel. We also investigate the role of artificial conductivity (AC). Although AC is necessary in the simulations to avoid "oily" features in the gas due to artificial surface tension, we fail to find any relation between using artificial conductivity and the appearance of seeded KH rolls. Including AC is necessary for the long-term behavior of the simulation (e.g. to get λ = 1/2, 1 KH rolls). In sensitive hydrodynamical simulations great care is however needed in selecting the AC signal velocity, with the default formulation leading to too much energy diffusion. We present new signal velocities that lead to less diffusion. The effects of the shock waves and of particle disorder become less important as the timescale of the physical problem (for the shearing layers problem: lower density contrast and higher Mach numbers) decreases. At the resolution of current galaxy formation simulations mixing is probably not important. However, mixing could become crucial for next-generation simulations.
Numerical modeling of Kelvin–Helmholtz instability using smoothed particle hydrodynamics
problem of an incompressible two-phase immiscible fluid in a stratified inviscid shear flow with interfacial tension. The time-dependent evolution of the two-fluid interface over a wide range of Richardson number (Ri) and for three different density ratios is numerically investigated. The simulation results are compared with analytical solutions in the linear regime. Having captured the physics behind KHI, the effects of gravity and surface tension on a two-dimensional shear layer are examined independently and together. It is shown that the growth rate of the KHI is mainly controlled by the value of the Ri number, not by the nature of the stabilizing forces. It was observed that the SPH method requires a Richardson number lower than unity (i.e. Ri ∼ = 0.8) for the onset of KHI, and that the artificial viscosity plays a significant role in obtaining physically correct simulation results that are in agreement with analytical solutions. The numerical algorithm presented in this work can easily handle two-phase fluid flow with various density ratios. NUMERICAL MODELING OF KELVIN-HELMHOLTZ INSTABILITY USING SPH Figure 7. Time evolution of the interface in the two-dimensional KHI problem for the density ratio of 2 / 1 = 10 at various Ri numbers: (a) t * = 0.5; (b) t * = 1.0; (c) t * = 1.5; and (d) t * = 2.0 ( = 0.001).
Numerical Simulations of the Kelvin–Helmholtz Instability with the Gadget-2 SPH Code
Environmental Science and Engineering, 2013
The method of Smoothed Particle Hydrodynamics (SPH) has been widely studied and implemented for a large variety of problems, ranging from astrophysics to fluid dynamics and elasticity problems in solids. However, the method is known to have several deficiencies and discrepancies in comparison with traditional meshbased codes. In particular, there has been a discussion about its ability to reproduce the Kelvin-Helmholtz Instability in shearing flows. Several authors reported that they were able to reproduce correctly the instability by introducing some improvements to the algorithm. In this contribution, we compare the results of the Read, Hayfield & Agertz (2010) implementation of the SPH algorithm with the original Gadget-2 N-body/SPH code.
The Astrophysical Journal, 2013
We use a suite of cosmological hydrodynamic simulations, run by two fixed grid codes, to investigate the properties of solenoidal and dilatational motions of the intergalactic medium (IGM), and the impact of numerical viscosity on turbulence in a LCDM universe. The codes differ only in the spatial difference discretization. We find that (1) The vortical motion grows rapidly since z = 2, and reaches ∼ 10km/s − 90km/s at z = 0. Meanwhile, the small-scale compressive ratio r CS drops from 0.84 to 0.47, indicating comparable vortical and compressive motions at present. (2) Power spectra of the solenoidal velocity possess two regimes, ∝ k −0.89 and ∝ k −2.02 , while the total and dilatational velocity follow the scaling k −1.88 and k −2.20 respectively in the turbulent range. The IGM turbulence may contain two distinct phases, the supersonic and post-supersonic phases. The non-thermal pressure support, measured by the vortical kinetic energy, is comparable with the thermal pressure for ρ b ≃ 10 − 100, or T < 10 5.5 K at z = 0.0. The deviation of the baryon fraction from the cosmic mean shows a preliminary positive correlation with the turbulence pressure support. (4) A relatively higher numerical viscosity would dissipate both the compressive and vortical motions of the IGM into thermal energy more effectively, resulting in less developed vorticity, remarkably shortened inertial range, and leading to nonnegligible uncertainty in the thermal history of gas accretion . Shocks in regions
Application of Molecular Hydrodynamics to Astrophysical Flows. II: Unconditional Stability
Progress of Theoretical Physics, 2008
We discuss the physical background ef the molecular hydrodynamics method (MH), a new computational fluid dynamics (CFD) technique that we proposed recently, and furt･her test it to simulate isothermal flows including those of zero temperature gas. The problems considered are a shock tube of an isothermal gas, a rotating cone test, a box shear flow test, and a Keplerian disc. We demonstrate that the MH is unconditionally stable in spite of the fact that the scheme is time-explicit. Because of this, we may choose any tirne step without losing stability. The only penalty for using a longer time step is a gradual degradation of the quality of the solutien.
Velocity shear induced phenomena in solar and astrophysical flows
2006
The present dissertation is a summary of the work that I have conducted in Centre for Plasma Astrophysics, (CPA, K.U.Leuven) and Abastumani Astrophysical Observatory (AbAO, Georgia). I would like to thank George Chagelishvili who introduced me into the subject of the presented study and for continuous support. Marcel Goossens was my supervisor in CPA and among many other things I would like to thank him for his patience during the last 5 years. Andria Rogava has led me in navigation to the Belgium. I thank him for help and support. At CPA I enjoyed hospitality for which I thank Stefaan Poedts and Arnold Debosscher. For friendly and stimulating atmosphere in the CPA I thank Anik De Groof, Jesse Andries, Bart van der Holst, Yuriy Voitenko and Jovo Vranjes who are presently in CPA, as well as Balazs