1,212 research outputs found
A matrix stability analysis of the carbuncle phenomenon
The carbuncle phenomenon is a shock instability mechanism which ruins all efforts to compute grid-aligned shock
waves using low-dissipative upwind schemes. The present study develops a stability analysis for two-dimensional steady shocks on structured meshes based on the matrix method. The numerical resolution of the corresponding eigenvalue problem confirms the typical odd–even form of the unstable mode and displays a Mach number threshold effect currently observed in computations. Furthermore, the present method indicates that the instability of steady shocks is not only governed by the upstream Mach number but also by the numerical shock structure. Finally, the source of the instability is localized in the upstream region, providing some clues to better understand and control the onset of the carbuncle
Low Mach number effect in simulation of high Mach number flow
In this note, we relate the two well-known difficulties of Godunov schemes:
the carbuncle phenomena in simulating high Mach number flow, and the inaccurate
pressure profile in simulating low Mach number flow. We introduced two simple
low-Mach-number modifications for the classical Roe flux to decrease the
difference between the acoustic and advection contributions of the numerical
dissipation. While the first modification increases the local numerical
dissipation, the second decreases it. The numerical tests on the double-Mach
reflection problem show that both modifications eliminate the kinked Mach stem
suffered by the original flux. These results suggest that, other than
insufficient numerical dissipation near the shock front, the carbuncle
phenomena is strongly relevant to the non-comparable acoustic and advection
contributions of the numerical dissipation produced by Godunov schemes due to
the low Mach number effect.Comment: 9 pages, 1 figur
A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws
We introduce a new methodology for adding localized, space-time smooth,
artificial viscosity to nonlinear systems of conservation laws which propagate
shock waves, rarefactions, and contact discontinuities, which we call the
-method. We shall focus our attention on the compressible Euler equations in
one space dimension. The novel feature of our approach involves the coupling of
a linear scalar reaction-diffusion equation to our system of conservation laws,
whose solution is the coefficient to an additional (and artificial)
term added to the flux, which determines the location, localization, and
strength of the artificial viscosity. Near shock discontinuities, is
large and localized, and transitions smoothly in space-time to zero away from
discontinuities. Our approach is a provably convergent, spacetime-regularized
variant of the original idea of Richtmeyer and Von Neumann, and is provided at
the level of the PDE, thus allowing a host of numerical discretization schemes
to be employed. We demonstrate the effectiveness of the -method with three
different numerical implementations and apply these to a collection of
classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the
Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a
classical continuous finite-element implementation using second-order
discretization in both space and time, FEM-C. Second, we use a simplified WENO
scheme within our -method framework, WENO-C. Third, we use WENO with the
Lax-Friedrichs flux together with the -equation, and call this WENO-LF-C.
All three schemes yield higher-order discretization strategies, which provide
sharp shock resolution with minimal overshoot and noise, and compare well with
higher-order WENO schemes that employ approximate Riemann solvers,
outperforming them for the difficult Leblanc shock tube experiment.Comment: 34 pages, 27 figure
Simulation of the growth of the 3D Rayleigh-Taylor instability in Supernova Remnants using an expanding reference frame
Context: The Rayleigh-Taylor instabilities generated by the deceleration of a
supernova remnant during the ejecta-dominated phase are known to produce
finger-like structures in the matter distribution which modify the geometry of
the remnant. The morphology of supernova remnants is also expected to be
modified when efficient particle acceleration occurs at their shocks. Aims: The
impact of the Rayleigh-Taylor instabilities from the ejecta-dominated to the
Sedov-Taylor phase is investigated over one octant of the supernova remnant. We
also study the effect of efficient particle acceleration at the forward shock
on the growth of the Rayleigh-Taylor instabilities. Methods: We modified the
Adaptive Mesh Refinement code RAMSES to study with hydrodynamic numerical
simulations the evolution of supernova remnants in the framework of an
expanding reference frame. The adiabatic index of a relativistic gas between
the forward shock and the contact discontinuity mimics the presence of
accelerated particles. Results: The great advantage of the super-comoving
coordinate system adopted here is that it minimizes numerical diffusion at the
contact discontinuity, since it is stationary with respect to the grid. We
propose an accurate expression for the growth of the Rayleigh-Taylor structures
that connects smoothly the early growth to the asymptotic self-similar
behaviour. Conclusions: The development of the Rayleigh-Taylor structures is
affected, although not drastically, if the blast wave is dominated by cosmic
rays. The amount of ejecta that makes it into the shocked interstellar medium
is smaller in the latter case. If acceleration occurs at both shocks the extent
of the Rayleigh-Taylor structures is similar but the reverse shock is strongly
perturbed.Comment: 15 pages, 12 figures, accepted for publication in Astronomy and
Astrophysics with minor editorial changes. Version with full resolution
images can be found at http://www.lpl.arizona.edu/~ffrasche/~12692.pd
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
- …