A solution to the overdamping problem when simulating dust-gas mixtures with smoothed particle hydrodynamics

We present a fix to the overdamping problem found by Laibe & Price (2012) when simulating strongly coupled dust-gas mixtures using two different sets of particles using smoothed particle hydrodynamics. Our solution is to compute the drag at the barycentre between gas and dust particle pairs when computing the drag force by reconstructing the velocity field, similar to the procedure in Godunov-type solvers. This fixes the overdamping problem at negligible computational cost, but with additional memory required to store velocity derivatives. We employ slope limiters to avoid spurious oscillations at shocks, finding the van Leer Monotonized Central limiter most effective.Comment: 6 pages, 5 figures, accepted to MNRA

Dusty gas with one fluid in smoothed particle hydrodynamics

In a companion paper we have shown how the equations describing gas and dust as two fluids coupled by a drag term can be reformulated to describe the system as a single fluid mixture. Here we present a numerical implementation of the one-fluid dusty gas algorithm using Smoothed Particle Hydrodynamics (SPH). The algorithm preserves the conservation properties of the SPH formalism. In particular, the total gas and dust mass, momentum, angular momentum and energy are all exactly conserved. Shock viscosity and conductivity terms are generalised to handle the two-phase mixture accordingly. The algorithm is benchmarked against a comprehensive suit of problems: dustybox, dustywave, dustyshock and dustyoscill, each of them addressing different properties of the method. We compare the performance of the one-fluid algorithm to the standard two-fluid approach. The one-fluid algorithm is found to solve both of the fundamental limitations of the two- fluid algorithm: it is no longer possible to concentrate dust below the resolution of the gas (they have the same resolution by definition), and the spatial resolution criterion h < csts, required in two-fluid codes to avoid over-damping of kinetic energy, is unnecessary. Implicit time stepping is straightforward. As a result, the algorithm is up to ten billion times more efficient for 3D simulations of small grains. Additional benefits include the use of half as many particles, a single kernel and fewer SPH interpolations. The only limitation is that it does not capture multi-streaming of dust in the limit of zero coupling, suggesting that in this case a hybrid approach may be required.Comment: Accepted for publication in MNRAS. Numerical code and input files for dustybox, wave and shock tests available from http://users.monash.edu.au/~dprice/ndspmhd

A fast and explicit algorithm for simulating the dynamics of small dust grains with smoothed particle hydrodynamics

We describe a simple method for simulating the dynamics of small grains in a dusty gas, relevant to micron-sized grains in the interstellar medium and grains of centimetre size and smaller in protoplanetary discs. The method involves solving one extra diffusion equation for the dust fraction in addition to the usual equations of hydrodynamics. This "diffusion approximation for dust" is valid when the dust stopping time is smaller than the computational timestep. We present a numerical implementation using Smoothed Particle Hydrodynamics (SPH) that is conservative, accurate and fast. It does not require any implicit timestepping and can be straightforwardly ported into existing 3D codes.Comment: 15 pages, 10 figures, accepted to MNRAS. Code implementation (ndspmhd v2.1) and setup of test problems available at: http://users.monash.edu.au/~dprice/ndspmhd/. v3: sign errors fixed as per erratum to published pape

Dust and gas mixtures with multiple grain species - a one-fluid approach

GL acknowledges funding from the European Research Council for the FP7 ERC advanced grant project ECOGAL. DJP is very grateful for funding via an ARC Future Fellowship, FT130100034, and Discovery Project grant DP130102078.We derive the single-fluid evolution equations describing a mixture made of a gas phase and an arbitrary number of dust phases, generalizing the approach developed by Laibe & Price. A generalization for continuous dust distributions as well as analytic approximations for strong drag regimes is also provided. This formalism lays the foundation for numerical simulations of dust populations in a wide range of astrophysical systems while avoiding limitations associated with a multiple-fluid treatment. The usefulness of the formalism is illustrated on a series of analytical problems, namely the DUSTYBOX, DUSTYSHOCK and DUSTYWAVE problems as well as the radial drift of grains and the streaming instability in protoplanetary discs. We find physical effects specific to the presence of several dust phases and multiple drag time-scales, including non-monotonic evolution of the differential velocity between phases and increased efficiency of the linear growth of the streaming instability. Interestingly, it is found that under certain conditions, large grains can migrate outwards in protoplanetary discs. This may explain the presence of small pebbles at several hundreds of astronomical units from their central star.Publisher PDFPeer reviewe

Dusty gas with one fluid

In this paper, we show how the two-fluid equations describing the evolution of a dust and gas mixture can be reformulated to describe a single fluid moving with the barycentric velocity of the mixture. This leads to evolution equations for the total density, momentum, the differential velocity between the dust and the gas phases and either the dust-to-gas ratio or the dust fraction. The equations are similar to the usual equations of gas dynamics, providing a convenient way to extend existing codes to simulate two-fluid mixtures without modifying the code architecture. Our approach avoids the inherent difficulties related to the standard approach where the two phases are separate and coupled via a drag term. In particular, the requirements of infinite spatial and temporal resolution as the stopping time tends to zero are no longer necessary. This means that both small and large grains can be straightforwardly treated with the same method, with no need for complicated implicit schemes. Since there is only one resolution scale the method also avoids the problem of unphysical trapping of one fluid (e.g. dust) below the resolution of the other. We also derive a simplified set of equations applicable to the case of strong drag/small grains, consisting of the standard fluid equations with a modified sound speed, plus an advection-diffusion equation for the dust-to-gas ratio. This provides a simple and fast way to evolve the mixture when the stopping time is smaller than the Courant timestep. We present a Smoothed Particle Hydrodynamics implementation in a companion paper.Comment: Accepted for publication in MNRAS (very minor revisions included

Is the dust-to-gas ratio constant in molecular clouds?

We perform numerical simulations of dusty, supersonic turbulence in molecular clouds. We model 0.1, 1 and 10 {\mu}m sized dust grains at an initial dust-to-gas mass ratio of 1:100, solving the equations of combined gas and dust dynamics where the dust is coupled to the gas through a drag term. We show that, for 0.1 and 1 {\mu}m grains, the dust-to-gas ratio deviates by typically 10-20% from the mean, since the stopping time of the dust due to gas drag is short compared to the dynamical time. Contrary to previous findings, we find no evidence for orders of magnitude fluctuation in the dust-to-gas ratio for 0.1 {\mu}m grains. Larger, 10 {\mu}m dust grains may have dust-to-gas ratios increased by up to an order of magnitude locally. Both small (0.1 {\mu}m) and large ($\gtrsim$ 1 {\mu}m) grains trace the large-scale morphology of the gas, however we find evidence for 'size-sorting' of grains, where turbulence preferentially concentrates larger grains into dense regions. Size-sorting may help to explain observations of 'coreshine' from dark clouds, and why extinction laws differ along lines of sight through molecular clouds in the Milky Way compared to the diffuse interstellar medium.Comment: 6 pages, 4 figures, accepted for publication in MNRAS Letters, videos available at https://www.youtube.com/channel/UC7J6IDzQklFzKV3c6pBqxU

Dusty gas with SPH - I. Algorithm and test suite

We present a new algorithm for simulating two-fluid gas and dust mixtures in Smoothed Particle Hydrodynamics (SPH), systematically addressing a number of key issues including the generalised SPH density estimate in multi-fluid systems, the consistent treatment of variable smoothing length terms, finite particle size, time step stability, thermal coupling terms and the choice of kernel and smoothing length used in the drag operator. We find that using double-hump shaped kernels improves the accuracy of the drag interpolation by a factor of several hundred compared to the use of standard SPH bell-shaped kernels, at no additional computational expense. In order to benchmark our algorithm, we have developed a comprehensive suite of standardised, simple test problems for gas and dust mixtures: dustybox, dustywave, dustyshock, dustysedov and dustydisc, the first three of which have known analytic solutions. We present the validation of our algorithm against all of these tests. In doing so, we show that the spatial resolution criterion \Delta < cs ts is a necessary condition in all gas+dust codes that becomes critical at high drag (i.e. small stopping time ts) in order to correctly predict the dynamics. Implicit timestepping and the implementation of realistic astrophysical drag regimes are addressed in a companion paper.Comment: Accepted for publication in MNRA