A direct Eulerian GRP scheme for relativistic hydrodynamics: One-dimensional case

The paper proposes a direct Eulerian generalized Riemann problem (GRP) scheme for one-dimensional relativistic hydrodynamics. It is an extension of the Eulerian GRP scheme for compressible non-relativistic hydrodynamics proposed in [M. Ben-Artzi, J.Q. Li, G. Warnecke, A direct Eulerian GRP scheme for compressible fluid flows, J. Comput. Phys. 218 (2006) 19–43]. Two main ingredients, the Riemann invariant and the Rankine–Hugoniot jump condition, are directly used to resolve the local GRP in the Eulerian formulation, and thus the crucial and delicate Lagrangian treatment in the original GRP scheme [3] can be avoided. Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed GRP scheme.     

A variational formulation of relativistic hydrodynamics

We combine Taub's and Ray's variational approaches to relativistic hydrodynamics of perfect fluids into another simple formulation.     

Robust particle methods for relativistic hydrodynamics

In this paper we describe how the equations of relativistic fluid dynamics can be solved numerically using the particle method SPH.     

Stable first-order particle-frame relativistic hydrodynamics for dissipative systems

We propose a stable first-order relativistic dissipative hydrodynamic equation in the particle frame (Eckart frame) for the first time. The equation to be proposed was in fact previously derived by the authors and a collaborator from the relativistic Boltzmann equation. We demonstrate that the equilibrium state is stable with respect to the time evolution described by our hydrodynamic equation in the particle frame. Our equation may be a proper starting point for constructing second-order causal relativistic hydrodynamics, to replace Eckart's particle-flow theory.     

Simple solutions of relativistic hydrodynamics for longitudinally expanding systems

Simple, self-similar, analytic solutions of 1+1-dimensional relativistic hydrodynamics are presented, generalizing Bjorken's solution to inhomogeneous rapidity distribution.     

Simple solutions of relativistic hydrodynamics for cylindrically symmetric systems

Simple, self-similar, analytic solutions of 1+3-dimensional relativistic hydrodynamics are presented for cylindrically symmetric fireballs corresponding to central collisions of heavy ions at relativistic bombarding energies.     

An adaptive GRP scheme for compressible fluid flows

This paper presents a second-order accurate adaptive generalized Riemann problem (GRP) scheme for one and two dimensional compressible fluid flows. The current scheme consists of two independent parts: Mesh redistribution and PDE evolution. The first part is an iterative procedure. In each iteration, mesh points are first redistributed, and then a conservative interpolation formula is used to calculate the cell-averages and the slopes of conservative variables on the resulting new mesh. The second part is to evolve the compressible fluid flows on a fixed nonuniform mesh with the Eulerian GRP scheme, which is directly extended to two-dimensional arbitrary quadrilateral meshes. Several numerical examples show that the current adaptive GRP scheme does not only improve the resolution as well as accuracy of numerical solutions with a few mesh points, but also reduces possible errors or oscillations effectively.     

Extending the reach of hydrodynamics for relativistic heavy ion collisions

Recent and ongoing improvements to hydrodynamic treatments at RHIC are extending the physics reach of hydrodynamics, and improving the phenomenology. Here, the links between technological improvements and the extension of physics are emphasized.     

Simple solutions of relativistic hydrodynamics for systems with ellipsoidal symmetry

Simple, self-similar, analytic solutions of (1+3)-dimensional relativistic hydrodynamics are presented for ellipsoidally symmetric finite fireballs corresponding to non-central collisions of heavy ions at relativistic bombarding energies. The hydrodynamical solutions are obtained for a new, general family of equations of state with the possibility of describing phase transitions.     

A super space-time form of the hydrodynamics of free relativistic fluids

The dynamics of a free, special relativistic, adiabatic fluid results from an odd dynamical system in an space-time.     

A Eulerian PDF scheme for LES of nonpremixed turbulent combustion with second-order accurate mixture fraction

Monte Carlo simulations of joint probability density function (PDF) approaches have been developed in the past largely with Reynolds averaged Navier Stokes (RANS) applications. Current interests are in the extension of PDF approaches to large eddy simulation (LES). As LES resolves accurately the large scales of turbulence in time, the Monte Carlo simulation and the flow field need to be tightly coupled. A tight coupling can be achieved if the consistency between the scalar field solution obtained via finite-volume (FV) methods and that from the stochastic solution of the PDF is ensured. For nonpremixed turbulent flames with two distinct streams, the local reactive mixture is described by the mixture fraction. A Eulerian Monte Carlo method is developed to achieve a second-order accuracy in the instantaneous filtered mixture fraction that is consistent with the corresponding FV. The performances of the proposed scheme are extensively evaluated using a one-dimensional model. Then, the scheme is applied to two cases with LES. The first one is a non-reacting mixing flow of two different fluids. The second case is the Sandia piloted turbulent flame D with a steady state flamelet model. Both results confirm the consistency of the proposed method to the level of filtered mixture fraction.     

Exact solutions of relativistic perfect fluid hydrodynamics for a QCD Equation of State

The half-life of the mirror $\beta$ decay of ³¹S has been measured at the IGISOL facility at the University of Jyv?skyl?. The value obtained is $T_{1/2}(^{31}S)=(2553.4\pm 1.8)$ ms, in agreement with previous measurements, but with a precision that is better by a factor of ten than the literature value previously adopted. When the new result is combined with the Q_EC value measured recently at JYFLTRAP, a precision of better than $10^{-3}$ is obtained for the ft value.     

Two-dimensional friction oscillator: group-preserving scheme and handy formulae

With vibration isolation of buildings and turbomachinery blades in mind, we study the dynamic behaviour of a single-mass two-degree-of-freedom oscillator with dry friction damper, viscous damper and elastic spring connected in parallel. The mass is mounted on an elastic supporting plate allowing movement in two directions on a plane. We formulate a multi-dimensional friction model, from which the sliding conditions and the sticking conditions of the mass are derived. For calculations we develop a group-preserving scheme, which preserves the projective proper orthochronous Lorentz group PSO_o(2,1) symmetry of the model in the sliding phase so as to satisfy automatically the sliding conditions at each time step without iteration at all. The oscillator is then subjected to simple harmonic excitations, and the responses are displayed. According to the simple harmonic balance method together with a circular orbit assumption on displacements, we derive closed-form formulae for handily estimating the steady state responses, which are then compared with the results calculated by the group-preserving scheme to confirm the applicability of the formulae. We also derive formulae specifically for a two-dimensional friction oscillator with rigid base support, which include an exact formula of the magnification factor and a simple formula for estimating the minimum driving force amplitude (or the maximum friction force bound) to avoid sticking.     

A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension

We describe a cell-centered Godunov scheme for Lagrangian gas dynamics on general unstructured meshes in arbitrary dimension. The construction of the scheme is based upon the definition of some geometric vectors which are defined on a moving mesh. The finite volume solver is node based and compatible with the mesh displacement. We also discuss boundary conditions. Numerical results on basic 3D tests problems show the efficiency of this approach. We also consider a quasi-incompressible test problem for which our nodal solver gives very good results if compared with other Godunov solvers. We briefly discuss the compatibility with ALE and/or AMR techniques at the end of this work. We detail the coefficients of the isoparametric element in the appendix.     

Implementation of the GRP scheme for computing radially symmetric compressible fluid flows

The study of radially symmetric compressible fluid flows is interesting both from the theoretical and numerical points of view. Spherical explosion and implosion in air, water and other media are well-known problems in application. Typical difficulties lie in the treatment of singularity in the geometrical source and the imposition of boundary conditions at the symmetric center, in addition to the resolution of classical discontinuities (shocks and contact discontinuities). In the present paper we present the implementation of direct generalized Riemann problem (GRP) scheme to resolve this issue. The scheme is obtained directly by the time integration of the fluid flows. Our new contribution is to show rigorously that the singularity is removable and derive the updating formulae for mass and energy at the center. Together with the vanishing of the momentum, we obtain new numerical boundary conditions at the center, which are then incorporated into the GRP scheme. The main ingredient is the passage from the Cartesian coordinates to the radially symmetric coordinates.