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Survey of numerical methods for compressible fluids
The finite difference methods of Godunov, Hyman, Lax-Wendroff (two-step), MacCormack, Rusanov, the upwind scheme, the hybrid scheme of Harten and Zwas, the antidiffusion method of Boris and Book, and the artificial compression method of Harten are compared with the random choice known as Glimm's method. The methods are used to integrate the one-dimensional equations of gas dynamics for an inviscid fluid. The results are compared and demonstrate that Glimm's method has several advantages. 16 figs., 4 tables
Pion Interferometry for a Granular Source of Quark-Gluon Plasma Droplets
We examine the two-pion interferometry for a granular source of quark-gluon
plasma droplets. The evolution of the droplets is described by relativistic
hydrodynamics with an equation of state suggested by lattice gauge results.
Pions are assumed to be emitted thermally from the droplets at the freeze-out
configuration characterized by a freeze-out temperature . We find that the
HBT radius decreases if the initial size of the droplets decreases.
On the other hand, depends on the droplet spatial distribution and
is relatively independent of the droplet size. It increases with an increase in
the width of the spatial distribution and the collective-expansion velocity of
the droplets. As a result, the value of can lie close to
for a granular quark-gluon plasma source. The granular model of the emitting
source may provide an explanation to the RHIC HBT puzzle and may lead to a new
insight into the dynamics of the quark-gluon plasma phase transition.Comment: 5 pages, 4 figure
An Euler Solver Based on Locally Adaptive Discrete Velocities
A new discrete-velocity model is presented to solve the three-dimensional
Euler equations. The velocities in the model are of an adaptive nature---both
the origin of the discrete-velocity space and the magnitudes of the
discrete-velocities are dependent on the local flow--- and are used in a finite
volume context. The numerical implementation of the model follows the
near-equilibrium flow method of Nadiga and Pullin [1] and results in a scheme
which is second order in space (in the smooth regions and between first and
second order at discontinuities) and second order in time. (The
three-dimensional code is included.) For one choice of the scaling between the
magnitude of the discrete-velocities and the local internal energy of the flow,
the method reduces to a flux-splitting scheme based on characteristics. As a
preliminary exercise, the result of the Sod shock-tube simulation is compared
to the exact solution.Comment: 17 pages including 2 figures and CMFortran code listing. All in one
postscript file (adv.ps) compressed and uuencoded (adv.uu). Name mail file
`adv.uu'. Edit so that `#!/bin/csh -f' is the first line of adv.uu On a unix
machine say `csh adv.uu'. On a non-unix machine: uudecode adv.uu; uncompress
adv.tar.Z; tar -xvf adv.ta
New Relativistic Effects in the Dynamics of Nonlinear Hydrodynamical Waves
In Newtonian and relativistic hydrodynamics the Riemann problem consists of
calculating the evolution of a fluid which is initially characterized by two
states having different values of uniform rest-mass density, pressure and
velocity. When the fluid is allowed to relax, one of three possible
wave-patterns is produced, corresponding to the propagation in opposite
directions of two nonlinear hydrodynamical waves. New effects emerge in a
special relativistic Riemann problem when velocities tangential to the initial
discontinuity surface are present. We show that a smooth transition from one
wave-pattern to another can be produced by varying the initial tangential
velocities while otherwise maintaining the initial states unmodified. These
special relativistic effects are produced by the coupling through the
relativistic Lorentz factors and do not have a Newtonian counterpart.Comment: 4 pages, 5 figure
Dust flow in gas disks in the presence of embedded planets
We study the dynamics of gas and dust in a protoplanetary disk in the
presence of embedded planets. We investigate the conditions for dust-gap
formation in terms of particle size and planetary mass. We also monitor the
amount of dust that is accreted by the planet relative to the amount of gas,
which is an important parameter in determining the enrichment of solids in
giant planets compared to the solid content of the central star. We use a new
two-fluid hydrodynamics code to solve the flow equations for both gas and dust.
For the gas, we use a Godunov-type scheme with an approximate Riemann solver
(the Roe solver). The dust is treated as a pressureless fluid by essentially
the same numerical method as is used for the gas. We find that it only takes a
planet of 0.05 Jupiter masses to open up a gap in a disk with a significant
population of mm-sized particles. Dust particles larger than 150 micron
participate in gap formation. We also find that the formation of the gap
severely slows down dust accretion compared to that in the gas. Therefore, it
is not possible to enrich a newly formed giant planet in solids, if these
solids are contained in particles with sizes from 150 micron to approximately
10 cm.Comment: 13 pages, 12 figures, accepted for publication in A&
RODEO: a new method for planet-disk interaction
In this paper we describe a new method for studying the hydrodynamical
problem of a planet embedded in a gaseous disk. We use a finite volume method
with an approximate Riemann solver (the Roe solver), together with a special
way to integrate the source terms. This new source term integration scheme
sheds new light on the Coriolis instability, and we show that our method does
not suffer from this instability. The first results on flow structure and gap
formation are presented, as well as accretion and migration rates. For Mpl <
0.1 M_J and Mpl > 1.0 M_J (M_J = Jupiter's mass) the accretion rates do not
depend sensitively on numerical parameters, and we find that within the disk's
lifetime a planet can grow to 3-4 M_J. In between these two limits numerics
play a major role, leading to differences of more than 50 % for different
numerical parameters. Migration rates are not affected by numerics at all as
long as the mass inside the Roche lobe is not considered. We can reproduce the
Type I and Type II migration for low-mass and high-mass planets, respectively,
and the fastest moving planet of 0.1 M_J has a migration time of only 2.0 10^4
yr.Comment: Accepted for publication in A&
Hydrodynamic simulations with the Godunov SPH
We present results based on an implementation of the Godunov Smoothed
Particle Hydrodynamics (GSPH), originally developed by Inutsuka (2002), in the
GADGET-3 hydrodynamic code. We first review the derivation of the GSPH
discretization of the equations of moment and energy conservation, starting
from the convolution of these equations with the interpolating kernel. The two
most important aspects of the numerical implementation of these equations are
(a) the appearance of fluid velocity and pressure obtained from the solution of
the Riemann problem between each pair of particles, and (b the absence of an
artificial viscosity term. We carry out three different controlled
hydrodynamical three-dimensional tests, namely the Sod shock tube, the
development of Kelvin-Helmholtz instabilities in a shear flow test, and the
"blob" test describing the evolution of a cold cloud moving against a hot wind.
The results of our tests confirm and extend in a number of aspects those
recently obtained by Cha (2010): (i) GSPH provides a much improved description
of contact discontinuities, with respect to SPH, thus avoiding the appearance
of spurious pressure forces; (ii) GSPH is able to follow the development of
gas-dynamical instabilities, such as the Kevin--Helmholtz and the
Rayleigh-Taylor ones; (iii) as a result, GSPH describes the development of curl
structures in the shear-flow test and the dissolution of the cold cloud in the
"blob" test.
We also discuss in detail the effect on the performances of GSPH of changing
different aspects of its implementation. The results of our tests demonstrate
that GSPH is in fact a highly promising hydrodynamic scheme, also to be coupled
to an N-body solver, for astrophysical and cosmological applications.
[abridged]Comment: 19 pages, 13 figures, MNRAS accepted, high resolution version can be
obtained at
http://adlibitum.oats.inaf.it/borgani/html/papers/gsph_hydrosim.pd
An approach for solving the boundary free edge difficulties in SPH modelling: application to a viscous accretion disc in close binaries
In this work, we propose a SPH interpolating Kernel reformulation suitable
also to treat free edge boundaries in the computational domain. Application to
both inviscid and viscous stationary low compressibility accretion disc models
in Close Binaries (CB) are shown. The investigation carried out in this paper
is a consequence of the fact that a low compressibility modelling is crucial to
check numerical reliability.
Results show that physical viscosity supports a well-bound accretion disc
formation, despite the low gas compressibility, when a Gaussian-derived Kernel
(from the Error Function) is assumed, in extended particle range - whose Half
Width at Half Maximum (HWHM) is fixed to a constant value - without any
spatial restrictions on its radial interaction (hereinafter GASPHER). At the
same time, GASPHER ensures adequate particle interpolations at the boundary
free edges. Both SPH and adaptive SPH (hereinafter ASPH) methods lack accuracy
if there are not constraints on the boundary conditions, in particular at the
edge of the particle envelope: Free Edge (FE) conditions. In SPH, an
inefficient particle interpolation involves a few neighbour particles; instead,
in the second case, non-physical effects involve both the boundary layer
particles themselves and the radial transport.
Either in a regime where FE conditions involve the computational domain, or
in a viscous fluid dynamics, or both, a GASPHER scheme can be rightly adopted
in such troublesome physical regimes. Despite the applied low compressibiity
condition, viscous GASPHER model shows clear spiral pattern profiles
demonstrating the better quality of results compared to SPH viscous ones.
Moreover a successful comparison of results concerning GASPHER 1D inviscid
shock tube with analytical solution is also reported.Comment: 18 pages, 12 figure
An approach to the Riemann problem in the light of a reformulation of the state equation for SPH inviscid ideal flows: a highlight on spiral hydrodynamics in accretion discs
In physically inviscid fluid dynamics, "shock capturing" methods adopt either
an artificial viscosity contribution or an appropriate Riemann solver
algorithm. These techniques are necessary to solve the strictly hyperbolic
Euler equations if flow discontinuities (the Riemann problem) are to be solved.
A necessary dissipation is normally used in such cases. An explicit artificial
viscosity contribution is normally adopted to smooth out spurious heating and
to treat transport phenomena. Such a treatment of inviscid flows is also widely
adopted in the Smooth Particle Hydrodynamics (SPH) finite volume free
Lagrangian scheme. In other cases, the intrinsic dissipation of Godunov-type
methods is implicitly useful. Instead "shock tracking" methods normally use the
Rankine-Hugoniot jump conditions to solve such problems. A simple, effective
solution of the Riemann problem in inviscid ideal gases is here proposed, based
on an empirical reformulation of the equation of state (EoS) in the Euler
equations in fluid dynamics, whose limit for a motionless gas coincides with
the classical EoS of ideal gases. The application of such an effective solution
to the Riemann problem excludes any dependence, in the transport phenomena, on
particle smoothing resolution length in non viscous SPH flows. Results on
1D shock tube tests, as well as examples of application for 2D turbulence and
2D shear flows are here shown. As an astrophysical application, a much better
identification of spiral structures in accretion discs in a close binary (CB),
as a result of this reformulation is also shown here.Comment: 19 pages, 17 figure
Three-dimensional CFD simulations with large displacement of the geometries using a connectivity-change moving mesh approach
This paper deals with three-dimensional (3D) numerical simulations involving 3D moving geometries with large displacements on unstructured meshes. Such simulations are of great value to industry, but remain very time-consuming. A robust moving mesh algorithm coupling an elasticity-like mesh deformation solution and mesh optimizations was proposed in previous works, which removes the need for global remeshing when performing large displacements. The optimizations, and in particular generalized edge/face swapping, preserve the initial quality of the mesh throughout the simulation. We propose to integrate an Arbitrary Lagrangian Eulerian compressible flow solver into this process to demonstrate its capabilities in a full CFD computation context. This solver relies on a local enforcement of the discrete geometric conservation law to preserve the order of accuracy of the time integration. The displacement of the geometries is either imposed, or driven by fluid–structure interaction (FSI). In the latter case, the six degrees of freedom approach for rigid bodies is considered. Finally, several 3D imposed-motion and FSI examples are given to validate the proposed approach, both in academic and industrial configurations
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